ICMAT€¦ · NONCOMMUTATIVE DE LEEUW THEOREMS MARTIJN CASPERS, JAVIER PARCET MATHILDE PERRIN AND...

NONCOMMUTATIVE DE LEEUW THEOREMS

MARTIJN CASPERS, JAVIER PARCET

MATHILDE PERRIN AND ERIC RICARD

Abstract. Let H be a subgroup of some locally compact group G. Assume

H is approximable by discrete subgroups and G admits neighborhood baseswhich are ‘almost-invariant’ under conjugation by finite subsets of H. Let

m : G → C be a bounded continuous symbol giving rise to an Lp-bounded

Fourier multiplier (not necessarily cb-bounded) on the group von Neumannalgebra of G for some 1 ≤ p ≤ ∞. Then, m|H yields an Lp-bounded Fourier

multiplier on the group von Neumann algebra of H provided the modularfunction ∆H coincides with ∆G over H. This is a noncommutative form of de

Leeuw’s restriction theorem for a large class of pairs (G,H), our assumptions

on H are quite natural and recover the classical result. The main differencewith de Leeuw’s original proof is that we replace dilations of gaussians by

other approximations of the identity for which certain new estimates on almost

multiplicative maps are crucial. Compactification via lattice approximationand periodization theorems are also investigated.

Introduction

In 1965, Karel de Leeuw proved three fundamental theorems for EuclideanFourier multipliers. Given a bounded continuous symbol m : Rn → C, let usconsider the corresponding multiplier

Tmf(ξ) = m(ξ)f(ξ),

Tmf(x) =

∫Rnm(ξ)f(ξ)e2πi〈x,ξ〉 dξ.

The main results in [13] may be stated as follows:

i) Restriction. If m is continuous and Tm is Lp(Rn)-bounded

Tm|H :

∫H

f(h)χh dµ(h) 7→∫

H

m(h)f(h)χh dµ(h)

extends to a Lp(H)-bounded multiplier for any subgroup H ⊂ Rn where theχh’s stand for the characters on the dual group and µ is the Haar measure.

ii) Periodization. Given H ⊂ Rn any closed subgroup and mq : Rn/H→ Cbounded, let mπ : Rn → C denote its H-periodization which is defined bymπ(ξ) = mq(ξ + H). Then we find∥∥Tmπ : Lp(Rn)→ Lp(Rn)

∥∥ =∥∥Tmq : Lp(Rn/H)→ Lp(Rn/H)

∥∥.iii) Compactification. Let Rnbohr be the Pontryagin dual of Rndisc equipped

with the discrete topology. Given m : Rn → C bounded and continuous, theLp(Rn)-boundedness of Tm is equivalent to the boundedness in Lp(Rnbohr)of the multiplier with the same symbol

Tm :∑

Rndisc

f(ξ)χξ 7→∑

Rndisc

m(ξ)f(ξ)χξ.

1

2 CASPERS, PARCET, PERRIN, RICARD

Together with Cotlar’s work [10], de Leeuw theorems may be regarded as the firstform of transference in harmonic analysis, prior to Calderon and Coifman/Weisscontributions [6, 8]. The combination of the above-mentioned results producesa large family of previously unknown Lp-bounded Fourier multipliers —a sampleof them will appear in Appendix A— and restriction/periodization are nowadaysvery well-known properties of Euclidean Fourier multipliers. Although not so muchknown, the compactification theorem was the core result of [13].

Our goal is to study these results within the context of general locally compactgroups. Shortly after de Leeuw, Saeki [48] extended these theorems to locallycompact abelian (LCA) groups with an approach which relies more on periodizationand the structure theorem of LCA groups. On the contrary, no analog transferenceresults in the frequency group seem to exist for nonabelian groups, see [14, 15, 26, 54]for a dual approach. This gap is partly justified by the noncommutative natureof the spaces involved. Namely, the action in de Leeuw theorems occurs in thefrequency groups and the Fourier multipliers must be defined in the correspondingduals. The dual of a nonabelian locally compact group can only be understoodas a quantum group whose underlying space is a noncommutative von Neumannalgebra. If µG denotes the left Haar measure on a locally compact group G andλG : G→ U(L2(G)) stands for the left regular representation on G, the group vonNeumann algebra LG is the weak-∗ closure in B(L2(G)) of operators of the form

f =

∫G

f(g)λG(g) dµG(g) with f ∈ Cc(G).

The Plancherel weight is determined by τG(f) = f(e) for f in Cc(G) ∗ Cc(G) and

Lp(G) denotes the noncommutative Lp space on (LG, τG). Although very naturalin operator algebra and noncommutative geometry, group von Neumann algebrasare not yet standard spaces in harmonic analysis. The early remarkable workof Cowling/Haagerup [12, 23] on approximation properties of these algebras wasperhaps the first contribution in the line of harmonic analysis. The Lp-theory wasnot seriously considered until [25]. However, only during very recent years a prolificseries of results have appeared in the literature [7, 31, 32, 33, 36, 40, 42].

In contrast with [13, 48] where compactification and periodization took the leadrespectively, we will first put the emphasis on restriction. Assume in what followsthat our groups are second countable. We say that a locally compact group His approximable by discrete subgroups (H ∈ ADS) when there exists a family oflattices (Γj)j≥1 in H and associated fundamental domains (Xj)j≥1 which form aneighborhood basis of the identity. On the other hand, we say that G has smallalmost-invariant neighborhoods with respect to H (G ∈ [SAIN]H) if for every F ⊂ Hfinite, there is a basis (Vj)j≥1 of symmetric neighborhoods of the identity with

limj→∞

µG

((h−1Vjh)4Vj

)µG(Vj)

= 0 for all h ∈ F.

Theorem A . Let H be a subgroup of some locally compact group G. AssumeH ∈ ADS and G ∈ [SAIN]H. Let m : G→ C be a bounded continuous symbol givingrise to an Lp-bounded multiplier for some 1 ≤ p ≤ ∞. Then∥∥Tm|H : Lp(H)→ Lp(H)

∥∥ ≤ ∥∥Tm : Lp(G)→ Lp(G)∥∥

provided the modular function ∆H coincides with the restriction of ∆G to H.

NONCOMMUTATIVE DE LEEUW THEOREMS 3

A natural difficulty for the proof of Theorem A comes from the fact that we onlyassume boundedness of our multipliers. Indeed, when G is amenable, cb-boundedanalogs easily follow from the recent transference results in [7, 40] between Fourierand Schur multipliers. It should however be noted that, for Lp-bounded multipliersor even for cb-bounded multipliers over nonamenable groups, our approach requiresa different strategy which does not rely on previously known techniques.

Pairs (G,H) satisfying Theorem A include restriction onto Heisenberg groupsand other classical nilpotent groups. In fact, amenable ADS subgroups of locallycompact groups with ∆H = ∆G|H

also fulfill the hypotheses. Other nonamenablepairs will be considered in the paper. Our assumptions are indeed natural forthis degree of generality. The condition G ∈ [SAIN]H has its roots in de Leeuw’soriginal argument. Although not explicitly mentioned, a key point in his proofis the use of an approximate identity intertwining with the Fourier multiplier. Inthe Euclidean setting of [13], this was naturally achieved by using dilations of thegaussian, which is fixed by the Fourier transform. In our general setting, the heatkernel must be replaced by other approximations and the SIN condition —smallinvariant neighborhoods, which have been studied in the literature— yields certainapproximations intertwining with the Fourier multiplier. Our jump from SIN tothe more flexible almost-invariant class SAIN requires a more functional analyticapproach which circumvents the technicalities required for a heat kernel approachin such a general setting. The crucial novelty are certain estimates for almostmultiplicative maps of independent interest. Surprisingly, our argument is equallysatisfactory and much cleaner. We will prove a limiting intertwining behavior ofour approximation of the identity as a consequence of the following result.

Theorem B. Let (M, τ) be a semifinite von Neumann algebra equipped with anormal semifinite faithful trace. Let T :M→M be a subunital positive map withτ T ≤ τ . Then, given any 1 ≤ p ≤ ∞ and x ∈ L+

2p(M)∥∥T (x)− T (√x)2∥∥

2p≤ 1

2

∥∥T (x2)− T (x)2∥∥ 1

2

p.

We will use Haagerup’s reduction method [24] to extend the implications ofTheorem B for type III von Neumann algebras. This will be the key subtle point inproving Theorem A for nonunimodular G. Theorem B seems to provide new insighteven in the commutative setting. Namely, arguing as in the proof of Theorem Awe can use Theorem B to control the frequency support of a fractional power of afunction in terms of the frequency support of the original function, up to certainsmall Lp correction terms, we refer to Remark 1.5 for further details.

Let us now go back to the other assumption in Theorem A. The ADS propertyof H was implicitly used in de Leeuw’s original argument and could be a naturallimitation for restriction of Fourier multipliers in this general setting, perhaps morepowerful tools could be used for nice Lie groups [53]. In our case, we will just provethe validity of Theorem A for discrete subgroups Γ of a locally compact group Gin the class [SAIN]Γ. Then, assuming H ∈ ADS is approximated by (Γj)j≥1, thecomplete statement follows from the inclusion

[SAIN]H ⊂⋂j≥1

[SAIN]Γj ,

and the following noncommutative form of Igari’s lattice approximation [28, 29].


Theorem C. If G ∈ ADS is approximated by (Γj)j≥1∥∥Tm : Lp(G)→ Lp(G)∥∥ ≤ sup

j≥1

∥∥Tm|Γj : Lp(Γj)→ Lp(Γj)∥∥

for any 1 ≤ p ≤ ∞ and any bounded symbol m : G→ C continuous µG–a.e.

Apart from arbitrary discrete groups and many LCA groups, other nontrivialexamples in the ADS class include again Heisenberg groups and other nilpotentgroups. Although Theorem C is not very surprising, its proof is certainly technicaland it becomes a key point in our compactification theorem. Let G be a locallycompact group and write Gdisc for the same group equipped with the discretetopology. Our next goal is to determine under which conditions∥∥Tm : Lp(G)→ Lp(G)

∥∥ ∼ ∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)∥∥

for bounded continuous symbols. Of course, we may not expect that such anequivalence holds for arbitrary locally compact groups, since this would mean thatFourier multiplier Lp theory reduces to the class of discrete group von Neumannalgebras. Note also that restriction in the pair (G,H) always holds when both groupalgebras admit Lp-compactification since restriction within the family of discretegroups follows by taking conditional expectations. This gives another evidencethat compactification only holds under additional assumptions. We finally considerthe periodization problem. Let H be a normal closed subgroup of some locallycompact group G. Consider any bounded symbol mq : G/H → C (not necessarilycontinuous) and construct the H-periodization given by mπ(g) = mq(gH). Theperiodization problem consists in giving conditions under which∥∥Tmπ : Lp(G)→ Lp(G)

∥∥p∼∥∥Tmq : Lp(G/H)→ Lp(G/H)

∥∥p.

Theorem D. Let G be a locally compact unimodular group and H a normal closedsubgroup of G. Let us consider a bounded continuous symbol m : G → C andlet mq : G/H → C be bounded with H-periodization mπ(g) = mq(gH). Then, thefollowing inequalities hold for 1 ≤ p ≤ ∞ :

i) If G is ADS∥∥Tm : Lp(G)→ Lp(G)∥∥ ≤ ∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥.ii) If Gdisc is amenable∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥ ≤ ∥∥Tm : Lp(G)→ Lp(G)∥∥.

iii) If G is LCA∥∥Tmπ : Lp(G)→ Lp(G)∥∥ ≤ ∥∥Tmq : Lp(G/H)→ Lp(G/H)

∥∥.iv) If H is compact∥∥Tmq : Lp(G/H)→ Lp(G/H)

∥∥ ≤ ∥∥Tmπ : Lp(G)→ Lp(G)∥∥.

The unimodularity of G seems crucial for compactification, given the fact thatGdisc is always unimodular. The ADS condition is certainly natural to controlFourier multipliers on G by the same ones defined on Gdisc. It is an interestingproblem to decide whether this assumption is in fact necessary. As we will see theamenability in ii) and the commutativity in iii) (which goes back to Saeki) are veryclose to optimal. The inequality in iv) also holds for nonunimodular G.


Our conditions above can be substantially relaxed for amenable groups in theassumption that our multipliers are completely bounded. This follows from thetransference results in [7, 40] between Fourier and Schur multipliers —which workin the cb-setting for amenable groups— together with an approximation result forSchur multipliers from [36]. In particular, de Leeuw theorems hold in full generalityin this context as we shall prove in the last section. The validity of our results fornonamenable groups is what forces us to find new arguments in this paper. We willclose this article with two appendices. In Appendix A we analyze a certain family ofidempotent Fourier multipliers in R. By using restriction and lattice approximationwe will relate these multipliers with Fefferman’s theorem for the ball [16] and solvea question from [32]. Appendix B contains an overview of what is known in thecontext of Jodeit’s multiplier theorem [29] for locally compact groups.

1. Almost multiplicative maps

In this section we shall prove Theorem B and some consequences of it whichwill be crucial in our approach to noncommutative restrictions. Our results are ofindependent interest in the context of almost multiplicative maps on Lp. Alongthis section (M, τ) will be a semifinite von Neumann algebra with a given normalsemifinite faithful trace. We will need the following classical inequalities. They arewell known for Schatten classes and can be found in Bhatia’s book [4, TheoremsIX.4.5 and X.1.1]. The proofs given there can be generalized to any semifinite vonNeumann algebra, but we will provide more direct arguments. The second resultis a one-sided generalization of the Powers-Størmer inequality.

Lemma 1.1. Given 1 ≤ p ≤ ∞, the identity

α12 γβ

12 =

1

2

∫Rα−is(αγ + γβ)βis

ds

cosh(πs)

holds in Lp(M) for any α, β, γ in L2p(M) with α, β ≥ 0. In particular

i)∥∥α 1

2 γβ12

∥∥p≤ 1

2

∥∥αγ + γβ∥∥p.

ii) If γ = γ∗,∥∥α 1

2 γα12

∥∥p≤∥∥αγ∥∥

p.

Proof. Inequalities i) and ii) follow from the first identity. By an approximation

argument we may assume that α12 and β

12 have discrete spectrum, so that they

are linear combinations of pairwise disjoint projections. By direct substitution thisreduces the problem to α, β ∈ R+ and γ = 1. Since the map z 7→ α1−zβz isholomorphic on the strip ∆ = 0 < Re z < 1 and continuous on its closure, itsvalue at z = 1

2 is given by the integral formula

α12 β

12 =

∫∂∆

α1−zβz dµ(z)

where µ is the harmonic measure on ∂∆ relative to the point z = 1/2. Now, sincethis measure gives the probability for a random walk from the point 1

2 of hitting theboundary ∂∆, it coincides at both components ∂j = Rez = j of the boundary(j = 0, 1). This means that there is a probability measure ν on R satisfying theidentity

α12 β

12 =

1

2

(∫Rα1−isβisdν(s) +

∫Rα−isβ1+is dν(s)

)=

1

2

∫Rα−is(α+ β)βis dν(s).


An inspection of the harmonic measure in ∂∆ yields dν(s) = ds/ cosh(πs). Indeed,this can be obtained from the harmonic measure on the unit circle by means of aconformal map, see for instance [3, p. 93]. The proof is complete.

Lemma 1.2. If p ≥ 1, 0 < θ ≤ 1 and x, y ∈ L+θp(M)∥∥xθ − yθ∥∥

p≤∥∥x− y∥∥θ

θp.

Proof. Cutting x and y by some of their spectral projections we may assume that(M, τ) is finite and x, y ∈M. We may also reduce the above estimate to the casex ≥ y ≥ 0. To that end, note that

‖a− b‖pp ≤ ‖a‖pp + ‖b‖ppfor a, b ≥ 0. Indeed, let q+ = 1a−b≥0 and q− = 1a−b<0 then

‖a− b‖pp = ‖q+(a− b)q+‖pp + ‖q−(b− a)q−‖pp≤ ‖q+aq+‖pp + ‖q−bq−‖pp ≤ ‖a‖pp + ‖b‖pp

as 0 ≤ q+(a−b)q+ ≤ q+aq+ and similarly for the other term. Now let δ+, δ− be thepositive and negative parts of δ = x− y = δ+ − δ−. Let us consider the operators

a = (x+ δ−)θ − xθ and b = (y + δ+)θ − yθ.

Since y + δ+ = x + δ− we deduce that xθ − yθ = b − a. Moreover, by operatormonotonicity of s 7→ sθ, a and b are positive. Then the result for x + δ− ≥ x ≥ 0and y + δ+ ≥ y ≥ 0 yields∥∥xθ − yθ∥∥p

p= ‖a− b‖pp ≤ ‖a‖pp + ‖b‖pp ≤ ‖δ−‖

θpθp + ‖δ+‖θpθp = ‖δ‖θpθp =

∥∥x− y∥∥θpθp.

Let us then prove the assertion when x ≥ y ≥ 0. We will also assume y ≥ ε1to avoid unnecessary technical complications. Using the integral representation fors ∈M invertible

sθ = cθ

∫R+

tθs

s+ t

dt

twith cθ =

(∫R+

uθ

u(1 + u)du)−1

.

Differentiating the above integral formula and putting δ = x− y, we get

xθ − yθ = cθ

∫ 1

0

∫R+

tθ(y + uδ + t)−1δ(y + uδ + t)−1 dt du.

Now, for a fixed u ∈ [0, 1], we consider the continuous function

t 7→ ut =1√θ

(y + uδ)1−θ

2

(y + uδ + t)

with positive values in the commutative algebra generated by y + uδ. Moreover

cθ

∫R+

tθu2t dt =

cθθ

∫R+

tθ(y + uδ)1−θ

(y + uδ + t)2dt =

cθθ

∫R+

tθ

(1 + t)2dt = 1.

Therefore, the map on M defined by

z 7→ cθ

∫R+

tθutzut dt


is unital, completely positive and trace preserving. In particular, it extends to acontraction on Lp(M) for all 1 ≤ p ≤ ∞, see [24, Remark 5.6] for further details.We deduce∥∥xθ − yθ∥∥

p≤ θ

∫ 1

0

∥∥(y + uδ) θ−1

2 δ(y + uδ) θ−1

2∥∥pdu

= θ

∫ 1

0

∥∥δ 12 (y + uδ

)θ−1δ

12

∥∥pdu ≤ θ

∫ 1

0

uθ−1‖δθ‖p du = ‖δ‖θθp,

where the last inequality follows from the operator monotonicity of s 7→ s1−θ.

Proof of Theorem B. Given 1 ≤ p ≤ ∞, we claim that

(1.1)∥∥R(x)−R(

√x)2∥∥

2p≤ 1

2

∥∥R(x2)−R(x)2∥∥ 1

2

p

for any subunital positive map R : `n∞ → M with values in M∩ L1(M) and anypositive x ∈ `n∞. The assumption above about the range of R is to ensure thatR : `n∞ → Lp(M) is well-defined. Before proving this claim, let us show how thisimplies the assertion. Indeed, if T : M → M is a subunital positive map withτ T ≤ τ it follows from [24, Remark 5.6] that T extends to a positive contractionon Lp(M). Now, when x ∈ L+

2p(M) has a finite spectrum x =∑nj=0 λjpj with

λ0 = 0 and pj spectral projections, then pj ∈ M ∩ L1(M) for j ≥ 1 and we maydefine R : `n∞ →M by R(ej) = T (pj), where (ej)

nj=1 denotes the canonical basis of

`n∞. The map R clearly satisfies the assumptions of our claim and R(zα) = T (xα)for z =

∑nj=1 λjej and any α > 0. Hence (1.1) gives∥∥T (x)− T (

√x)2∥∥

2p≤ 1

2

∥∥T (x2)− T (x)2∥∥ 1

2

p

as desired. The general case x ∈ L+2p(M) follows by standard approximations. Let

xn =

n2∑k=1

k

n1[ kn ,

k+1n )(x).

It is an exercise to show that for α ∈ 1, 2, 12, x

αn → xα in the appropriate Lq-space.

Let us now prove the claim (1.1). As usual, (eij)ni,j=1 will denote the canonical

basis of the matrix algebra Mn. We first use an explicit Stinespring’s decompositionfor R. Let π : `n∞ →Mn be the usual diagonal inclusion and put

u∗ =

n∑j=1

ej1 ⊗R(ej)12 ∈Mn,1(M),

so that we have R(x) = uπ(x)u∗. As R is subunital uu∗ ≤ 1M and u∗u ≤ 1Mn(M).For any positive y ∈ `n∞, we get

R(y2)−R(y)2 = uπ(y)(1− u∗u

)π(y)u∗ =

∣∣√1− u∗uπ(y)u∗∣∣2.

Let us consider the following operators

a = R(√x) = uπ(

√x)u∗ ∈ M,

b =√

1− u∗uπ(√x)√

1− u∗u ∈ Mn(M).


Then we find z ∈ Mn,1(M) with ‖z‖∞ ≤ 1 so that√

1− u∗uπ(√x)u∗ = b

12 za

12

and√

1− u∗uπ(x)u∗ =√

1− u∗uπ(√x)((1−u∗u)+u∗u

)π(√x)u∗ = b

32 za

12 +b

12 za

32 .

We apply Lemma 1.1 twice to conclude. First∥∥R(x)−R(√x)2∥∥

2p=∥∥b 1

2 za12

∥∥2

4p=∥∥a 1

2 z∗bza12

∥∥2p≤∥∥az∗bz∥∥

2p≤∥∥az∗b∥∥

2p.

Then, taking (α, γ, β) = (a, a12 z∗b

12 , b) we obtain∥∥az∗b∥∥

2p≤ 1

2

∥∥a 32 z∗b

12 + a

12 z∗b

32

∥∥2p

=1

2

∥∥R(x2)−R(x)2∥∥ 1

2

p.

This completes the proof of our claim and also the proof of Theorem B.

Corollary 1.3. Let T :M→M be a subunital positive map with τ T ≤ τ . Thenthere exists a universal constant C > 0 such that the following inequality holds forany x ∈ L+

2 (M) and any 0 < θ ≤ 1∥∥T (xθ)− xθ∥∥

2θ

≤ C∥∥T (x)− x

∥∥ θ22‖x‖

θ22 .

Proof. Given x ∈ L+2 (M), note that

‖T (x)− x‖22 ≤ ‖Tx‖22 + ‖x‖22 ≤ τ(T (x2)) + ‖x‖22 ≤ 2 ‖x‖22by Kadison’s inequality for T and τ T ≤ τ . In particular, the result is triviallytrue for θ = 1 with constant 2

14 . We claim the assertion holds for θ = 2−n with

constant 32 . We will proceed by induction since we know it holds for n = 0. By

Lemma 1.2 and n+ 1 applications of Theorem B∥∥T (x2−(n+1)

)− x2−(n+1)∥∥2

2n+2

≤∥∥T (x2−(n+1)

)2 − x2−n∥∥

2n+1

≤∥∥T (x2−(n+1)

)2 − T (x2−n)∥∥

2n+1 +∥∥T (x2−n)− x2−n

∥∥2n+1

≤n∏j=0

2−2−j

︸︷︷︸Cn=22−n−2

∥∥T (x2)− T (x)2∥∥2−(n+1)

1+∥∥T (x2−n)− x2−n

∥∥2n+1 .

On the other hand, Kadison’s inequality and τ T ≤ τ yield∥∥T (x2)− T (x)2∥∥

1= τ

(T (x2)− T (x)2

)≤ τ

(x2 − T (x)2

)≤

∥∥x2 − T (x)2∥∥

1≤ 2

∥∥T (x)− x∥∥

2‖x‖2

since T extends to a contraction on L2(M) by [24, Remark 5.6]. Combining theabove estimates with the induction hypothesis for θ = 2−n we finally deduce that∥∥T (x2−(n+1)

)− x2−(n+1)∥∥2

2n+2 ≤[3

2+ 22−(n+1)

Cn

]‖x‖2

−(n+1)

2

∥∥T (x)− x∥∥2−(n+1)

2.

However, the constant in the right hand side is less than 9/4 and the result followsfor θ = 2−(n+1) which completes the induction argument. For other values of0 < θ < 1 we write θ = α2−(n+1) for some α ∈ (1, 2). Recall that s 7→ sα isoperator convex and s 7→ s

α2 is operator concave on R+, so that

T(x2−(n+1))α ≤ T

(xθ)≤ T

(x2−n

)α2 .


In conjunction with Lemma 1.2, Theorem B and our result for θ = 2−n, this yields∥∥T (xθ)− xθ∥∥

2θ

≤∥∥T (xθ)− T (x2−n)

α2

∥∥2θ

+∥∥T (x2−n)

α2 − xθ

∥∥2θ

≤∥∥T (x2−n)

α2 − T (x2−(n+1)

)α∥∥

2θ

+∥∥T (x2−n)

α2 − xθ

∥∥2θ

≤∥∥T (x2−n)− T (x2−(n+1)

)2∥∥α2

2n+1 +∥∥T (x2−n)− x2−n

∥∥α22n+1

≤[(

22−(n+1)

Cn

)α2

+(3

2

)α2]∥∥Tx− x∥∥ θ2

2‖x‖

θ22 .

Hence, a simple calculation shows that the result follows for some C ≤ 3+√

22 .

Corollary 1.4. Let T :M→M be a subunital positive map with τ T ≤ τ . Thenthere exists a universal constant C > 0 such that the following inequality holds forany self-adjoint y ∈ L2(M) with polar decomposition y = u|y| and any 0 < θ ≤ 1∥∥T (u|y|θ)− u|y|θ

∥∥2θ

≤ C∥∥T (y)− y

∥∥ θ42‖y‖

3θ4

2 .

Proof. Let us write y = y+ − y− for the decomposition of y into its positive andnegative parts, so that u|y|θ = yθ+ − yθ−. By positivity of the trace and T , we have

τ(T (y+)y+

)+ τ(T (y−)y−

)≥ τ

(T (y)y

).

In particular∥∥T (y+)− y+

∥∥2

2+∥∥T (y−)− y−

∥∥2

2≤ 2‖y‖22 − 2τ

(T (y)y

)≤ 2

∥∥T (y)− y∥∥

2‖y‖2.

Using this and Corollary 1.3 we deduce∥∥T (u|y|θ)− u|y|θ∥∥ 4θ2θ

≤[∥∥T (yθ+)− yθ+

∥∥2θ

+∥∥T (yθ−)− yθ−

∥∥2θ

] 4θ

≤ 24θ−1[∥∥T (yθ+)− yθ+

∥∥ 4θ2θ

+∥∥T (yθ−)− yθ−

∥∥ 4θ2θ

]≤ (2C)

4θ

2

[∥∥T (y+)− y+

∥∥2

2‖y+‖22 +

∥∥T (y−)− y−∥∥2

2‖y−‖22

]≤ (2C)

4θ

2

[∥∥T (y+)− y+

∥∥2

2+∥∥T (y−)− y−

∥∥2

2

]‖y‖22

≤ (2C)4θ

∥∥T (y)− y∥∥

2‖y‖32.

The assertion follows taking powers θ/4 at both sides of the above estimate.

Remark 1.5. The above corollary will be useful to localize the frequency support

of square roots for elements in Lp(G). This is even interesting in the commutativecase where we may control the frequency support of a fractional power fθ in termsof the frequency support of f , up to small Lp corrections. As an illustration, ifthe Fourier transform of f ∈ L2(R) is supported by (−α, α), we may consider thepositive definite functions

ζβ(x) =(

1− |x|2β

)+

for β > 0.


The associated Fourier multipliers are positive, unital and trace preserving, so thatwe are in position to apply our results above. When β = α/2ε, we obtain thatsupp ζβ ⊂ 1

ε (−α, α) and 1− ζβ(x) ≤ ε for |x| ≤ α. This yields for p ≥ 2∥∥∥(ζβ(f2p )∧ − (f

2p )∧)∨∥∥∥p

p≤ C

∥∥ζβ f − f ∥∥2‖f‖2 . ε‖f‖22.

Remark 1.6. It is well known that if T :M→M satisfies the above hypothesisand τ is finite, then its fixed points form a ∗-subalgebra. This is not true anymorewhen τ is semifinite, take for instance the map x 7→ s∗xs on B(`2) where s isa one-sided shift. Nevertheless, in general, it is not difficult to show using thegeneralized singular value decomposition that if x ∈ L+

1 (M) ∪ L+2 (M) satisfies

T (x) = x, then T (xθ) = xθ for θ ∈ [0, 1]. Hence one could think of an ultraproductargument to get perturbation results as given explicitly in Corollary 1.3, with anupper bound of the form F (‖T (x)−x‖2) for certain continuous function F vanishingat 0. Unfortunately, semifiniteness is not preserved by ultraproduct and one wouldhave to deal with type III von Neumann algebras. The situation is then much moreintricate (even to define T on Lp(M)), that is why we choose to deduce the typeIII result from the semifinite one in Section 7.2. The fact that there exists a unitalcompletely positive map T : (M, ϕ) → (M, ϕ) with ϕ T = ϕ but T does notcommute with the modular group of ϕ (think of a right multiplier on a quantumgroup with its left Haar measure) is an evidence that in the type III situation oneneed extra arguments as those of Corollary 7.4.

2. Group algebras

Let G be a locally compact group equipped with its left Haar measure µG. LetλG : G → B(L2(G)) be the left regular representation λG(g)(ξ)(h) = ξ(g−1h)for any ξ ∈ L2(G). When no confusion can arise, we shall write µ, λ for the leftHaar measure and the left regular representation of G. Recall the definition of theconvolution in G

ξ ∗ η(g) =

∫G

ξ(h)η(h−1g) dµ(h).

We say that ξ ∈ L2(G) is left bounded if the map η ∈ Cc(G) 7→ ξ ∗ η ∈ L2(G)extends to a bounded operator on L2(G), denoted by λ(ξ). This operator definesthe Fourier transform of ξ. The weak operator closure of the linear span of λ(G)defines the group von Neumann algebra LG. It can also be described as the weakclosure in B(L2(G)) of operators of the form

f =

∫G

f(g)λ(g) dµ(g) = λ(f ) with f ∈ Cc(G).

The Plancherel weight τG : LG+ → [0,∞] is determined by the identity

τG(f∗f) =

∫G

|f(g)|2 dµ(g)

when f = λ(f ) for some left bounded f ∈ L2(G) and τG(f∗f) = ∞ for any otherf ∈ LG. Again, we shall just write τ for τG when the underlying group is clearfrom the context. After breaking into positive parts, this extends to a weight ona weak-∗ dense domain within the algebra LG. It will be instrumental to observethat the standard identity

τ(f) = f(e)


applies for f = λ(f ) ∈ λ(Cc(G)∗Cc(G)), see [44, Section 7.2] and [50, Section VII.3]for a detailed construction of the Plancherel weight. Note that for G discrete, τcoincides with the natural finite trace given by τ(f) = 〈fδe, δe〉. It is clear thatthe Plancherel weight is tracial if and only if G is unimodular, which will be thecase until Section 7. In the unimodular case, (LG, τ) is a semifinite von Neumannalgebra and we may construct the noncommutative Lp-spaces

Lp(LG, τ) = Lp(G) =

λ(Cc(G) ∗ Cc(G))

‖ ‖pfor 1 ≤ p < 2

λ(Cc(G))‖ ‖p

for 2 ≤ p <∞,

where the norm is given by

‖f‖p = τ(|f |p)1/p

and the p-th power is calculated by functional calculus applied to the (possiblyunbounded) operator f , we refer to Pisier/Xu’s survey [47] for more details onnoncommutative Lp-spaces. On the other hand, since left bounded functions aredense in L2(G), the map λ : ξ 7→ λ(ξ) extends to an isometry from L2(G) to

L2(G). We will refer to it as the Plancherel isometry and use it repeatedly in thesequel with no further reference. Given a symbol m : G→ C, we may consider theassociated multiplier Tm defined by

Tm(f) =

∫G

m(g)f(g)λ(g) dµ(g) for f ∈ Cc(G) ∗ Cc(G).

Tm is called an Lp-Fourier multiplier if it extends to a bounded map on Lp(G).

The rest of this section will be devoted to collect some elementary results aroundamenability and Fell absorption principles that will be used in the sequel. We willalso need the following result, which we prove for completeness.

Lemma 2.1. Let G be a second countable locally compact unimodular group. Thenthe group von Neumann algebra LH is a von Neumann subalgebra of LG for anyclosed unimodular subgroup H of G.

Proof. By the Effros-Mackey cross section theorem [49, Theorem 5.4.2], thereexists a Borel measurable map σ : H\G → G defined on the space of right cosetsof G. Hence, we have a Borel measurable correspondence between G and H\G×Hgiven by

Υ : G 3 g 7→ (Hg, h(g)) ∈ H\G×H,

where g = h(g)σ(Hg). According to [18, Theorem 2.49] for right cosets and sinceboth G and H are unimodular, we know that there exists a G-invariant Radonmeasure on right cosets. Therefore, the map

ξ 7→ ξ Υ−1

defines an isometry between L2(G) and L2(H\G × H). This allows us to identifythe algebras B(L2(G)) and B(L2(H\G)⊗2 L2(H)). Then, for any h ∈ H we get theidentity (

id⊗ λH(h))(ξ Υ−1)(Hg, h(g)) = ξ(h−1h(g)σ(Hg))

= ξ(h−1g) = λG(h)(ξ)(g)

for ξ ∈ L2(G) and g ∈ G, which proves that LH ' λG(h) : h ∈ H′′ ⊂ LG.


In the sequel, if no confusion is possible and when Lemma 2.1 applies, we mightuse the notation λ(h) to denote both λG(h) and λH(h). Let us now recall somewell-known characterizations of amenability. Recall that amenability is stable underclosed subgroups, quotients, direct products and group extensions. After that, wealso give a formulation of Fell absorption principle in Lp from [41].

Lemma 2.2. TFAE for any locally compact group G:

i) G is amenable

ii) Følner condition. Given ε > 0 and F ⊂ G finite, there exists UF,ε ⊂ G offinite positive measure such that µ(UF,εg4 UF,ε) < εµ(UF,ε) for all g ∈ F.

iii) Almost invariant vectors. Given ε > 0 and F ⊂ G finite, there exists anorm one function ξ ∈ L2(G) such that ‖λ(g)ξ− ξ‖L2(G) < ε for all g ∈ F.

iv) The inequality∥∥∥∑g∈F

ag

∥∥∥M≤∥∥∥∑g∈F

ag ⊗ λ(g)∥∥∥M⊗LG

holds for any finite F ⊂ G, any von Neumann algebraM and (ag)g∈F ⊂M.

Lemma 2.3. Given a discrete group G, we have:

i) If π : G→ U(H) is strongly continuous, then

λ⊗ π ' λ⊗ 1H

are unitarily equivalent with 1H the trivial representation on H.

ii) Let π : G→ U(H) be strongly continuous and assume N = π(G)′′ is finite.Then, given 1 ≤ p ≤ ∞, any semifinite von Neumann algebra M and anya : G→ Lp(M) continuous and compactly supported we have∥∥∥∫

G

a(g)⊗ λ(g)⊗ π(g) dµ(g)∥∥∥Lp(M⊗LG⊗N )

=∥∥∥ ∫

G

a(g)⊗ λ(g) dµ(g)∥∥∥Lp(M⊗LG)

.

3. Lattice approximation

In this section, we want to deduce the boundedness of an Lp-Fourier multiplierfrom the uniform boundedness of its restriction to certain families of lattices. Asstated in Theorem C, this will be possible if G is approximated by these latticesin the sense G ∈ ADS defined in the Introduction. Observe that if G ∈ ADSis approximated by (Γj)j≥1, then the union ∪jΓj of the approximating lattices isdense in G. Indeed, let g ∈ G and V be an open neighborhood of g. Then V g−1

is an open neighborhood of e and for j large enough we have Xj ⊂ V g−1. Let gjbe the representant of g−1 in Xj . In other words, there exists γj ∈ Γj such thatgj = γjg

−1. This implies γj = gjg ∈ Xjg ⊂ V , so that Γj ∩ V 6= ∅ and we deducethe density result mentioned above. In the proof of Theorem C we shall need acouple of auxiliary results, which are stated below.

Lemma 3.1. If G admits a lattice Γ with ∆G|Γ= ∆Γ, then G is unimodular.


The result above implies that every ADS group is by definition unimodular. Inparticular, our preliminaries on group von Neumann algebras from Section 2 sufficefor Theorem C. We need one more elementary result.

Lemma 3.2. Let G be a locally compact group and K ⊂ Ω ⊂ G with K compactand Ω open. Let (Vj)j≥1 be a basis of neighborhoods of the identity. Then, thereexists an index j0 ≥ 1 such that for any j ≥ j0

K ⊂⋃g∈K

gVj ⊂ Ω.

Proof. Take a left invariant distance d on G, so that d(K,Ωc) = δ > 0. Sincediam(Vj)→ 0, any j0 with diam(Vj) < δ for j ≥ j0 satisfies the conclusion.

Proof of Theorem C. The case p = 2 is straightforward since m is continuousalmost everywhere and the union of lattices Γj is dense in G, so that the L∞-normof m is determined by lattice approximation. On the other hand, by a standardduality argument, we may assume that p < 2. Moreover, the case p = 1 followsfrom the assertion for 1 < p < 2 and the three lines lemma

‖Tm‖1→1 ≤ limp→1‖Tm‖p→p ≤ lim

p→1supj≥1‖Tm|Γj ‖p→p ≤ sup

j≥1‖Tm|Γj ‖1→1.

Therefore, we may and will assume in what follows that 1 < p < 2. The strategywill be to approximate Tmf weakly in Lp by a sequence Sjf constructed from afamily (Sj)j≥1 of uniformly bounded maps as follows. For each j ≥ 1 we first definethe map

Φj : LΓj 3 λ(γ) 7→ h∗jλ(γ)hj ∈ LG,

where hj = λ(1Xj ) ∈ LG. Since G is locally compact and (Xj)j≥1 is a basis ofneighborhoods of e, we may assume that Xj lies in a compact set. In particular wehave 0 < µ(Xj) <∞. It is clear that Φj is completely positive and we may definethe family of operators

Φpj = µ(Xj)−2+ 1

pΦj .

Now we note the straightforward inequality

‖Φ∞j (1)‖LG = µ(Xj)−2‖hj‖2LG ≤ µ(Xj)

−2‖1Xj‖2L1(G) = 1.

Moreover, since the sets (γXj)γ∈Γj are disjoint, we also have for γ ∈ Γj

τ(Φj(λ(γ))

)= τ

(h∗jλ(γ)hj

)=⟨λ(γ)hj , hj

⟩L2(G)

=⟨1γXj ,1Xj

⟩L2(G)

= µ(Xj)δγ,e = µ(Xj)τ(λ(γ)).

By complete positivity of Φj , the first estimate implies that Φ∞j : LΓj → LG is

a contractive map. The second estimate implies that Φ1j is trace preserving and

hence defines a contraction L1(LΓj) → L1(LG) by means of [24, Remark 5.6].Using interpolation of analytic families of operators, we get∥∥Φpj : Lp(Γj)→ Lp(G)

∥∥ ≤ 1 for 1 ≤ p ≤ ∞.

On the other hand, the L2-adjoints Ψj = Φ∗j are given by

Ψj(f) =∑γ∈Γj

τ(h∗jλ(γ−1)hjf

)λ(γ)


for f ∈ LG. Moreover, given 1 ≤ p ≤ ∞, consider the contractions

Ψpj = (Φp

′

j )∗ = µ(Xj)−1− 1

pΨj : Lp(G)→ Lp(Γj),

where p′ denotes the conjugate of p. We are finally ready to introduce the maps

Sj = ΦpjTm|ΓjΨpj = µ(Xj)

−3ΦjTm|ΓjΨj : Lp(G)→ Lp(G),

which are uniformly bounded by

Cp := supj≥1

∥∥Tm|Γj : Lp(Γj)→ Lp(Γj)∥∥.

If we fix f ∈ λ(Cc(G)∗Cc(G)), the sequence (Sjf)j≥1 is uniformly bounded in Lp(G)by Cp‖f‖p and it accumulates in the weak topology. We claim that Sjf weaklyconverges to Tmf = w-Lp- limj Sjf . The theorem will follow by the Lp-density ofλ(Cc(G) ∗ Cc(G)). To prove it, we can reduce ourselves to show

(3.1) Tmf = L2- limj→∞

Sjf for any f ∈ λ(Cc(G) ∗ Cc(G)).

Indeed, if it holds true and q is any τ -finite projection

limj→∞

∥∥qTmf − qSjf∥∥p ≤ ‖q‖r limj→∞

‖Tmf − Sjf‖2 = 0,

where 1p = 1

r + 12 . Hence

qTmf = Lp- limj→∞

(qSjf) = w-Lp- limj→∞

(qSjf) = q(w-Lp- lim

j→∞Sjf

)for any τ -finite projection q, which implies Tmf = w-Lp- limj Sjf . We now turn tothe proof of the key result (3.1). Let us introduce some notations. For j ≥ 1 define

Lj : L2(G) 3 f 7→ µ(Xj)−1hjf ∈ L2(G)

and

Pj : L2(G) 3 f 7→ 1

µ(Xj)

∑γ∈Γj

⟨f, λ(γ)hj

⟩L2(G)

λ(γ)hj ∈ L2(G).

Given g ∈ G and since (γXj)γ∈Γj forms a partition of G, there exists a uniqueγ ∈ Γj such that g ∈ γXj . Let us write γj(g) for this element and consider the mapmj : G→ C given by mj(g) = m(γj(g)). We claim that

i) Sj = L∗jPjTmjLj on L2(G).

ii) Lj , L∗j , Pj : L2(G)→ L2(G) are contractive and∥∥Tmj : L2(G)→ L2(G)

∥∥ ≤ ‖m‖∞.iii) Given f ∈ λ(Cc(G)), the following identity holds

limj→∞

∥∥Ljf − f∥∥2+∥∥L∗jf − f∥∥2

+∥∥Pjf − f∥∥2

+∥∥Tmjf − Tmf∥∥2

= 0.

In fact, the first three summands also converge to 0 for f ∈ L2(G).

The L2-convergence (3.1) follows from this. Indeed, i) gives for f ∈ λ(Cc(G))∥∥Tmf − Sjf∥∥2≤

∥∥Tmf − L∗jTmf∥∥2

+∥∥L∗jTmf − L∗jPjTmf∥∥2

+∥∥L∗jPjTmf − L∗jPjTmjf∥∥2

+∥∥L∗jPjTmjf − L∗jPjTmjLjf∥∥2

,


which clearly tends to 0 as j → ∞ by ii) and iii). Therefore, it suffices to justifythe properties i), ii) and iii). Let us start by noticing the following identity whichfollows from Plancherel’s isometry⟨

Tmjf, λ(γ)hj⟩L2(G)

=⟨mj f ,1γXj

⟩L2(G)

= m(γ)⟨f, λ(γ)hj

⟩L2(G)

.

Applying this to hjf we get

Sjf = µ(Xj)−3∑γ∈Γj

m(γ)⟨f, h∗jλ(γ)hj

⟩L2(G)

h∗jλ(γ)hj

= µ(Xj)−3∑γ∈Γj

⟨Tmj (hjf), λ(γ)hj

⟩L2(G)

h∗jλ(γ)hj = L∗jPjTmjLjf,

which proves i). Claim ii) for Lj follows from ‖µ(Xj)−1hj‖∞ ≤ ‖µ(Xj)

−11Xj‖1 ≤ 1.The boundedness for Pj is clear since it is the orthogonal projection onto the closedlinear span of (λ(γ)hj)γ∈Γj . The last assertion in ii) is trivial since ‖mj‖∞ ≤ ‖m‖∞.Let us finally prove the convergence results in property iii). By [18, Proposition

2.42], the family hj = µ(Xj)−11Xj forms an approximation of the identity, so that

limj→∞

∥∥hj ∗ ξ − ξ∥∥L2(G)= 0 for ξ ∈ L2(G).

By Plancherel’s isometry, this yields vanishing limits for the first two terms in iii).Moreover, the third term will converge to 0 if and only if the orthogonal projection

Pj of L2(G) onto span1γXj : γ ∈ Γj satisfies

(3.2) limj→∞

∥∥Pjξ − ξ∥∥2= 0 for any ξ ∈ L2(G).

By the density of the simple functions in L2(G), we may assume that ξ = 1Ω fora Borel subset Ω of G with finite measure. Moreover, since the Haar measure isouter regular, Ω can be assumed to be open. On the other hand, given any ε > 0and since µ is inner regular on open sets, there exists a compact set K ⊂ Ω suchthat µ(Ω \ K) ≤ ε. By applying Lemma 3.2 to the basis of neighborhoods of theidentity given by (X−1

j Xj)j≥1, we obtain that there exists j0 ≥ 1 such that for anyj ≥ j0

K ⊂⋃g∈K

γj(g)Xj ⊂⋃g∈K

gX−1j Xj ⊂ Ω.

The sets (γXj)γ∈Γj being disjoint, we can find a subset F ⊂ K satisfying

K ⊂⊔g∈F

γj(g)Xj ⊂ Ω.

Moreover, since the sets Ω and γj(g)Xj are of finite measure, the set F has to befinite. Hence, the function η =

∑g∈F 1γj(g)Xj satisfies ‖ξ − η‖22 ≤ µ(Ω \ K) ≤ ε

and the limit (3.2) is proved. It remains to consider the last term in iii). Let ε > 0and f ∈ λ(Cc(G)) be frequency supported by a compact set K. Since γj(g)→ g asj →∞ and m is continuous µ–a.e., we have mj → m µ-a.e. Moreover, by Egoroff’stheorem [19, Theorem 2.33], there exists a set E ⊂ K with µ(E) < ε and such thatmj → m uniformly on K \ E. Pick j0 ≥ 1 satisfying

supg∈K\E

|mj(g)−m(g)| ≤ ε 12


for all j ≥ j0. Then we get for j ≥ j0∥∥Tmjf − Tmf∥∥2

L2(G)=∥∥(mj −m)f

∥∥2

L2(G)

≤∫

K\E|f(g)|2

∣∣mj(g)−m(g)∣∣2dµ(g) +

∫E

|f(g)|2∣∣mj(g)−m(g)

∣∣2dµ(g)

which is dominated by ε(‖f‖22 + 4‖m‖2∞‖f‖2∞

)and proves iii) for the last term.

Remark 3.3. Modifying the proof above, we may extend Theorem C. Namely, ifG ∈ ADS is approximated by (Γj)j≥1 and both m : G → C and mj : G → C area.e. continuous symbols such that mj → m uniformly, then∥∥Tm : Lp(G)→ Lp(G)

∥∥ ≤ supj≥1

∥∥T(mj)|Γj: Lp(Γj)→ Lp(Γj)

∥∥for any 1 ≤ p ≤ ∞. Indeed, it suffices to define

Sj = µ(Xj)−3ΦjT(mj)|Γj

Ψj

and consider mj(g) = mj(γj(g)) in the proof of Theorem C. Then the fourthsummand in iii) will follow by means of Plancherel’s isometry noticing that wehave |mj(g) −m(g)| ≤ ‖mj −m‖∞ + |m(γj(g)) −m(g)| and hence converges to 0a.e. Then we conclude as in the proof of Theorem C.

Remark 3.4. We have not performed an extensive study of groups satisfying theADS condition. Apart from discrete groups and many LCA groups, of particularinterest to us is the Heisenberg group, defined as the set Hn = Rn × Rn × R withinner law (a, b, c) · (a′, b′, c′) = (a + a′, b + b′, c + c′ + 1

2 (〈a, b′〉 − 〈a′, b〉)). It is asimple example of ADS group. Namely, take for instance the family of latticesΓj = 1

jZn× 1

jZn× 1

2j2Z which trivially satisfy the ADS condition. Other nilpotent

groups satisfying the ADS condition are the groups H(K, n) of upper triangularmatrices over the field K = R,C with 1’s on the diagonal. In this case, a simplechoice of lattices is Γj = Idn + 〈jr−sZ⊗ er,s : 1 ≤ r < s ≤ n〉.

4. The restriction theorem

In this section we prove Theorem A for unimodular groups. In other words, weprove that under the SAIN condition, Lp-Fourier multipliers of a unimodular groupG restrict to multipliers of any ADS subgroup H, and this restriction map is normdecreasing. All our work so far will be needed in the proof.

Proof of Theorem A: Unimodular case. Let us first reduce the proof to theparticular case of discrete subgroups. Indeed, let H ∈ ADS approximable by thefamily (Γj)j≥1 and assume that G ∈ [SAIN]H. Since Γj ⊂ H ⊂ G and

[SAIN]H ⊂⋂j≥1

[SAIN]Γj ,

the pairs (G,Γj)j≥1 are under the hypotheses of Theorem A for discrete subgroups.Moreover, since H is ADS, it follows from Lemma 3.1 that H must be unimodular.Therefore, Theorem A for discrete subgroups in conjunction with Theorem C yields∥∥Tm|H : Lp(H)→ Lp(H)

∥∥≤ sup

j≥1

∥∥Tm|Γj : Lp(Γj)→ Lp(Γj)∥∥ ≤ ∥∥Tm : Lp(G)→ Lp(G)

∥∥.


Hence, we shall consider in what follows a discrete subgroup Γ of a locally compactunimodular group G satisfying G ∈ [SAIN]Γ. Observe that by unimodularity of Gand discreteness of Γ our assumption ∆G|Γ = ∆Γ is superfluous. We have reducedthe proof of Theorem A for G unimodular to this particular case. Arguing as wedid at the beginning of the proof of Theorem C and using the continuity of thesymbol for p = 2, it suffices to consider 2 < p < ∞. Moreover, by density of the

trigonometric polynomials in Lp(Γ), it is enough to prove that

(4.1) ‖Tm|Γ f‖Lp(Γ) ≤∥∥Tm : Lp(G)→ Lp(G)

∥∥ ‖f‖Lp(Γ)

for any trigonometric polynomial f ∈ LΓ. As we explained in the Introduction, thebasic idea is to construct an approximation of the identity in G which intertwineswith the pair (Tm, Tm|Γ ) in the limit. Let us fix such a trigonometric polynomialf0 ∈ LΓ and let F ⊂ Γ denote its frequency support

f0 =∑γ∈F

f0(γ)λ(γ).

Let (Vj)j≥1 be the symmetric neighborhood basis of the identity associated to F bythe SAIN condition. Moreover, since Γ is discrete, we may take j large enough andassume that the sets (γVj)γ∈Γ are disjoint. Let us define the selfadjoint elements

hj = µ(Vj)−1/2λ(1Vj ) with polar decomposition hj = uj |hj |, and set

Φqj : λ(γ) 7→ λ(γ)uj |hj |2q for γ ∈ Γ and 2 ≤ q ≤ ∞.

Then the proof will rely on the two following results.

Claim A. Given 2 ≤ q ≤ ∞, we have:

i) Φqj extends to a contraction Lq(Γ)→ Lq(G).

ii) Given any f ∈ LΓ frequency supported by F, we have

limj→∞

‖Φqjf‖Lq(G) = ‖f‖Lq(Γ).

Claim B. Given 2 ≤ q < p and any trigonometric polynomial f in LΓ

limj→∞

‖Φqj(Tm|Γ f)− Tm(Φqjf)‖Lq(G) = 0.

Let us finish the proof of Theorem A before proving these two claims. Let f0 bethe trigonometric polynomial in LΓ frequency supported by F that we have fixedabove. The algebra LΓ being finite, we have

‖Tm|Γ f0‖Lp(Γ) = limqp‖Tm|Γ f0‖Lq(Γ).

Since Tm|Γ f0 is also frequency supported by F, Claims A and B yield

‖Tm|Γ f0‖Lp(Γ) = limqp

limj→∞

‖Φqj(Tm|Γ f0)‖Lq(G)

= limqp

limj→∞

‖Tm(Φqjf0)‖Lq(G)

≤ limqp

limj→∞

∥∥Tm : Lq(G)→ Lq(G)∥∥‖Φqjf0‖Lq(G)

= limqp‖Tm‖q→q‖f0‖Lq(Γ) ≤

∥∥Tm : Lp(G)→ Lp(G)∥∥‖f0‖Lp(Γ).

The latter inequality follows by interpolation since

‖Tm‖q→q ≤ ‖Tm‖1−θ2→2‖Tm‖θp→p


for 1q = 1−θ

2 + θp , so that θ → 1 as q → p. This completes the proof of (4.1).

Proof of Claim A. Since the SAIN condition implies G second countable, we mayconsider LΓ as a von Neumann subalgebra of LG by Lemma 2.1. Thus Claim A i)is clear for q = ∞ by writing ‖Φ∞j f‖LG = ‖fuj‖LG ≤ ‖f‖LG = ‖f‖LΓ. Moreover,by Plancherel’s isometry and disjointness of the sets (γVj)γ∈Γ we get

(4.2) ‖Φ2jf‖2L2(G)

= ‖fhj‖2L2(G)= µ(Vj)

−1∥∥∥∑γ∈Γ

f(γ)1γVj∥∥∥2

L2(G)= ‖f‖2

L2(Γ).

Claim A i) then follows using interpolation for analytic families of operators, weleave the details to the reader. The upper estimate in Claim A ii) follows from i)and it suffices to show that limj ‖Φqj(f)‖q ≥ ‖f‖q for trigonometric polynomials fin LΓ frequency supported by F. Let q∗ be the L2-conjugate index of q, so that1/q + 1/q∗ = 1/2. We have

‖f‖Lq(Γ) = ‖f∗‖Lq(Γ) = sup‖k‖Lq∗ (Γ)≤1

k trigonometric polynomial

‖kf∗‖L2(Γ).

Fix such a polynomial k =∑γ∈M k(γ)λ(γ). Then, since Φ2

j is an isometry by (4.2)

‖kf∗‖L2(Γ) = ‖Φ2j (kf

∗)‖L2(G) = ‖kf∗hj‖L2(G)

≤ ‖kf∗hj − Φq∗

j (k)ujΦqj(f)∗‖L2(G) + ‖Φq

∗

j (k)ujΦqj(f)∗‖L2(G).

By Holder’s inequality and Claim A i)

‖Φq∗

j (k)ujΦqj(f)∗‖L2(G) ≤ ‖Φ

q∗

j (k)‖Lq∗ (G)‖Φqj(f)∗‖Lq(G) ≤ ‖Φ

qj(f)‖Lq(G).

For the first summand, let us prove that

limj→∞

‖kf∗hj − Φq∗

j (k)ujΦqj(f)∗‖L2(G) = 0.

This will complete the proof of Claim A. Since hj is self-adjoint

Φq∗

j (k)ujΦqj(f)∗ = kuj |hj |2/q

∗uj |hj |2/qu∗jf∗ = khjf

∗.

Then, by Plancherel’s isometry and the Cauchy-Schwarz inequality we get

‖kf∗hj − khjf∗‖L2(G)

=∥∥∥ ∑γ′∈M,γ∈F

k(γ′)f(γ)(λ(γ′)λ(γ−1)hj − λ(γ′)hjλ(γ−1)

)∥∥∥L2(G)

≤∑

γ′∈M,γ∈F

∣∣k(γ′) f(γ)∣∣ ∥∥λ(γ′)λ(γ−1)hj − λ(γ′)hjλ(γ−1)

∥∥L2(G)

=∑

γ′∈M,γ∈F

∣∣k(γ′) f(γ)∣∣µ(Vj)

−1/2∥∥1γ−1Vjγ − 1Vj

∥∥L2(G)

=∑

γ′∈M,γ∈F

∣∣k(γ′) f(γ)∣∣(µ(γ−1Vjγ4Vj)

µ(Vj)

) 12

≤( ∑γ′∈M,γ∈F

|k(γ′)f(γ)|2) 1

2( ∑γ′∈M,γ∈F

µ(γ−1Vjγ4Vj)µ(Vj)

) 12


= ‖k‖L2(G)‖f‖L2(G)|M|12

(∑γ∈F

µ(γ−1Vjγ4Vj)µ(Vj)

) 12

,

which converges to 0 as j →∞ since we assumed have that G ∈ [SAIN]Γ.

Proof of Claim B. Without loss of generality we may assume that f = λ(γ) for

some γ ∈ Γ by the triangle inequality in Lq(G). Replacing m by m(γ ·) we mayassume that γ = e (we leave the details to the reader here). This means that weare reduced to prove

(4.3) limj→∞

∥∥m(e)uj |hj |2q − Tm(uj |hj |

2q )∥∥Lq(G)

= 0.

Given ε > 0 and since m is continuous in e ∈ G, there exists a neighborhood Uεof the identity such that |m(g) − m(e)| < ε for every g ∈ Uε. Since G is locallycompact we may assume that Uε is relatively compact and so µ(Uε) <∞. Let Wε

be a symmetric neighborhood of e with W 2ε ⊂ Uε and define

ζ(g) =µ(Wε ∩ gWε)

µ(Wε)=〈λ(g)1Wε

,1Wε〉

µ(Wε).

Hence, ζ is a coefficient function of the left regular representation and the coefficientis given by the positive vector state with respect to the vector µ(Wε)

− 12 1Wε

. Itis then standard that ζ is continuous, positive definite and ζ(e) = 1. Furthermoreby construction supp ζ ⊂ Uε. Let Tζ be the associated Fourier multiplier, thenTζ : LG→ LG is a normal, trace preserving, unital, completely positive map. Thisimplies that it extends to a contraction

Tζ : Lp(G)→ Lp(G)

for every 1 ≤ p ≤ ∞. By Plancherel isometry we have∥∥Tζhj − hj∥∥2

L2(G)=∥∥(ζ − 1)µ(Vj)

− 12 1Vj

∥∥2

L2(G)=

1

µ(Vj)

∫Vj

|ζ(g)− 1|2dµ(g),

which converges to 0 as j → ∞ since Vj → e and ζ is continuous at e. Atthis point we need our result on almost multiplicative maps. Indeed, since hj is aself-adjoint operator of L2-norm one, we deduce from Corollary 1.4 that

(4.4) limj→∞

∥∥Tζ(uj |hj | 2q )− uj |hj |2q

∥∥Lq(G)

= 0.

Let us now prove (4.3). Setting zj = uj |hj |2/q we write

‖m(e)zj − Tmzj‖Lq(G) ≤ ‖m(e)(zj − Tζzj)‖Lq(G)

+ ‖m(e)Tζzj − Tm(Tζzj)‖Lq(G)

+ ‖Tm(Tζzj)− Tmzj‖Lq(G) = Aj +Bj + Cj .

By (4.4), limj Aj = limj Cj = 0. By definition of Uε we have∥∥T(m(e)−m)ζ : L2(G)→ L2(G)∥∥ = ‖(m(e)−m)ζ‖L∞(G) < ε.

On the other hand, since ‖Tζ : Lp(G)→ Lp(G)‖ = 1 we get∥∥T(m(e)−m)ζ : Lp(G)→ Lp(G)∥∥ ≤ |m(e)|+

∥∥Tm : Lp(G)→ Lp(G)∥∥.

Applying the three lines lemma to the symbol (m(e)−m)ζ we obtain

Bj ≤ ε1−θ(|m(e)|+

∥∥Tm : Lp(G)→ Lp(G)∥∥)θ for

1

q=

1− θ2

+θ

p.


This implies (4.3), which gives Claim B and completes the proof of Theorem A.

We end this section by giving some examples of groups satisfying the conditionsof Theorem A. We have already considered the ADS condition in the previoussection, so let us analyze the SAIN condition. There are two general conditionswhich imply small almost invariant neighborhoods:

• G ∈ [SIN]H (small invariant neighborhoods) if there exists a neighborhoodbasis of the identity of G consisting of open sets that are invariant underconjugation with respect to H, see for instance [22, 38] for this class of pairs(G,H) when G = H. Of course, we have [SIN]H ⊂ [SAIN]H.

• Another interesting class of pairs satisfying the SAIN condition is given byamenable discrete subgroups Γ satisfying ∆G|Γ

= ∆Γ, see Theorem 7.11below. As a consequence of it, we shall show that Theorem A holds forpairs (G,H) with H any ADS amenable group.

Concrete examples (even in the nonunimodular setting) will be given in Section 7.

Remark 4.1. Both properties above are strictly weaker than the SAIN conditionsince none of them is included in the other one. To see this, let us constructexamples of pairs (G,Γ), where Γ is a discrete subgroup of a unimodular, locallycompact group G, satisfying only one of these two properties:

i) The free group with two generators F2 can be represented as a (non-closed)subgroup of SO(3). This way F2 acts on R3 and the open balls Br(0) ⊂ R3

with center 0 and radius r are invariant under the action of F2. We mayconsider the semidirect product G = R3 o F2, which is unimodular sincethe action of F2 is measure preserving. Then the sets Br(0) are naturallycontained in G and in fact form a basis of neighborhood of the identity whichare invariant under conjugation with respect to F2. Hence R3oF2 ∈ [SIN]F2

but F2 is not amenable.

ii) Let G be the Heisenberg group in Rn and Γ = Zn × 0 × 0 ⊂ G. ThenΓ satisfies our second property above but G /∈ [SIN]Γ. Indeed, let U be asmall neighborhood of (0, 0, 0) invariant under conjugation by Γ. Assumethat U ⊂ Rn × Rn × [−L,L] for some L > 0. Since conjugation in theHeisenberg group gives

(−a,−b,−c) · (x, y, t) · (a, b, c) =(x, y, t− 〈a, y〉+ 〈b, x〉

),

we deduce that (x, y, t) ∈ U ⇒ (x, y, t − ay) ∈ U for all a ∈ Zn. But wecan find an element (x, y, t) ∈ U with y 6= 0 and a sequence (aj)j≥1 in Znverifying |ajy| → ∞ which contradicts this property.

Remark 4.2. We already know from Lemma 3.1 that every ADS group must beunimodular. On the other hand, it also holds that G ∈ [SAIN]H with ∆H = ∆G|Himplies H unimodular since

∆H(h) = ∆G(h) = limj→∞

µ(h−1Vjh)

µ(Vj)

= limj→∞

µ(h−1Vjh)− µ(h−1Vjh \ Vj)µ(Vj)


= limj→∞

µ(h−1Vjh ∩ Vj)µ(Vj)

= 1− limj→∞

µ(Vj \ h−1Vjh)

µ(Vj)= 1

for every h ∈ H. In particular, all our conditions in Theorem A point to theunimodularity of H. As we shall see in Section 8, this is not the case when we workwith amenable groups in the category of operator spaces. We leave as an openproblem to decide whether unimodularity is an essential assumption for restrictionof Fourier multipliers.

5. The compactification theorem

We now extend de Leeuw’s compactification theorem. In other words, given alocally compact group G, let us write Gdisc to denote the same group equippedwith the discrete topology. Under the conditions in Theorem D, we prove that theLp-boundedness of a Fourier multiplier on G is equivalent to the Lp-boundednessof that multiplier defined on Gdisc. In this section and for the sake of clarity, wewill write λ = λG and λ′ = λGdisc

for the left regular representation on G andGdisc respectively. Moreover, we shall use a similar terminology for trigonometricpolynomials in both LG and LGdisc

f =∑g∈F

f(g)λ(g) ⇔ f ′ =∑g∈F

f(g)λ′(g).

Before proving the compactification theorem, let us first discuss the conditions onthe group G that we impose. In de Leeuw’s proof of the compactification theoremfor Rn, the following basic properties were crucial:

P1) We have

Rn =⋃j≥1

2−jZn.

P2) There is an injective homomorphism Ψ : Rn → Rnbohr —the dual to thecanonical inclusion map Rndisc → Rn— with dense image and such thatf = f ′ Ψ for any pair (f, f ′) ∈ L∞(Rn) × L∞(Rnbohr) of trigonometricpolynomials with matching Fourier coefficients. In particular

‖f ′‖L∞(Rnbohr)= supξ∈Rn

|f ′ Ψ(ξ)| = supξ∈Rn

|f(ξ)| = ‖f‖L∞(Rn).

Of course, we will replace P1) by our ADS condition. On the other hand, P2)is not a general property of locally compact groups. Indeed, according to Lemma2.2 iv) for M = C (see the proof), if ‖f‖LG = ‖f ′‖LGdisc

for any trigonometricpolynomial f in LG then the amenability of G is equivalent to the amenability ofGdisc. However, this is false in general. Consider for instance the group G = SO(3)which is compact, hence amenable. On the contrary, since the free group F2 is asubgroup of Gdisc = SO(3)disc, the discretized group Gdisc is not amenable. In thefollowing result we show that ‖f‖LG = ‖f ′‖LGdisc

when Gdisc is amenable.

Lemma 5.1. If f is a trigonometric polynomial in LG:

i) We always have ‖f ′‖LGdisc≤ ‖f‖LG.

ii) The reverse inequality holds true whenever Gdisc is amenable.


Proof. Let (Vj)j≥1 be a symmetric basis of neighborhoods of the identity in G

and let F ⊂ G be finite. Then for j ≥ 1 large enough and hj = µ(Vj)−1/2λ(1Vj )

the following map is isometric

(5.1) Lhj : `2(F) 3 (ag)g∈F 7→(∑g∈F

agλ(g))hj ∈ L2(G).

Indeed, since (gVj)g∈F are disjoint for j large enough

‖Lhj (a)‖22 = µ(Vj)−1∥∥∥∑g∈F

agλ(1gVj )∥∥∥2

2=∑g∈F

|ag|2 = ‖a‖2`2(F).

To prove i), we first write

‖f ′‖LGdisc= sup

⟨f ′ξ1, ξ2

⟩`2(Gdisc)

where the supremum runs over all finite subset X ⊂ G and all ξ1, ξ2 ∈ `2(X) with‖ξ1‖2 = ‖ξ2‖2 = 1. Pick any such X and ξ1, ξ2. Since f ′ξ1 is supported by FX theinner product above can be taken in `2(S), where S = FX∪X. Applying (5.1) to this

finite set S, we may find an isometry Lh : `2(S)→ L2(G). Since Lh(f ′ξ1) = fLh(ξ1)⟨f ′ξ1, ξ2

⟩`2(S)

=⟨Lh(f ′ξ1), Lh(ξ2)

⟩L2(G)

=⟨fLh(ξ1), Lh(ξ2)

⟩L2(G)

≤ ‖f‖LG.

Taking suprema we obtain i). If Gdisc is amenable, Lemma 2.2 yields

‖f‖LG =∥∥∥∑g∈F

f(g)λ(g)∥∥∥LG

≤∥∥∥∑g∈F

f(g)λ(g)⊗ λ′(g)∥∥∥LG⊗LGdisc

= ‖f ′‖LGdisc,

where the last equality comes from Fell’s absorption principle in Lemma 2.3 ii).

Remark 5.2. It follows that Gdisc amenable ⇒ G amenable, but not reciprocally.

We can now prove Theorem D i) and ii), the noncommutative version of deLeeuw’s compactification theorem. The first implication requires P1) and followseasily from the lattice approximation in Theorem C. The second one requires ananalogue of P2) —Gdisc amenable, as suggested by Lemma 5.1— and it follows byadapting our restriction argument in Theorem A.

Proof of Theorem D i) and ii). If G ∈ ADS is approximated by lattices (Γj)j≥1,then Γj ⊂ Gdisc for j ≥ 1. Since both groups are discrete, we may restrict by takinga conditional expectation. In conjunction with Theorem C, we obtain∥∥Tm : Lp(G)→ Lp(G)

∥∥ ≤ supj≥1

∥∥Tm|Γj: Lp(Γj)→ Lp(Γj)∥∥

≤∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥.This proves i). For the converse implication, we may and will assume as in theproof of Theorem A that 2 < p <∞. Now, since Gdisc is amenable, we claim thatG ∈ [SAIN]Gdisc

. Namely, it follows by the exact same argument as in Theorem7.11 since our proof there does not use the fact that the topology on the subgroupis induced by the topology of G. Once we know that the SAIN condition holds, thegoal is to show that

‖Tmf ′‖Lp(Gdisc)≤∥∥Tm : Lp(G)→ Lp(G)

∥∥ ‖f ′‖Lp(Gdisc)


for any trigonometric polynomial f ′ ∈ LGdisc. Fix such a trigonometric polynomial

f ′0 =∑γ∈F f

′0(γ)λ′(γ) ∈ LG and let F ⊂ G denote its frequency support. Let

(Vj)j≥1 be the neighborhood basis of the identity associated to F by the SAIN

condition. Following the proof of Theorem A, define hj = µ(Vj)−1/2λ(1Vj ) with

polar decomposition hj = uj |hj |. The main difference with the restriction theoremis that we may no longer assume that the sets (gVj)g∈G are disjoint. Then wecannot define properly the maps Φpj for all j, since they are not contractive anylonger. However, this still holds true at the limit.

Claim A’. Let 2 ≤ q ≤ ∞. Then

i) If f ′ ∈ LGdisc is any trigonometric polynomial

limj→∞

∥∥fuj |hj | 2q ∥∥Lq(G)≤ ‖f ′‖

Lq(Gdisc).

ii) If f ′ ∈ LGdisc is frequency supported by F, we also have

limj→∞

∥∥fuj |hj | 2q ∥∥Lq(G)= ‖f ′‖

Lq(Gdisc).

The intertwining result we gave in Claim B of the proof of Theorem A —restatedconveniently without using the maps Φpj— holds replacing Γ by Gdisc with verbatim

the same argument. Moreover, Theorem D ii) follows from it and Claim A’ aboveexactly as in the proof of Theorem A. Thus, it suffices to justify this claim.

Proof of Claim A’. Let ε > 0 and let f ′ be any trigonometric polynomial inLGdisc. Since interpolation cannot be used any longer in our case, Claim A’ i) willsimply follow from the three-lines lemma. Let a = a(f ′, ε, q) be a trigonometricpolynomial in LG such that

(5.2)∥∥|f | q2 − a∥∥LG

=∥∥|f ′| q2 − a′∥∥LGdisc

<1

2εq/2,

where the equality comes from Lemma 5.1 (together with a standard approximationargument in the weak-∗ topology) since Gdisc is amenable, and a′ denotes thetrigonometric polynomial in LGdisc associated to a. By (5.1), there exists an indexj0 = j0(f ′, ε, q) such that

(5.3) ‖ahj‖L2(G) = ‖a′‖L2(Gdisc)

for any j ≥ j0.

The map Fj(z) = u|f |qz/2uj |hj |z —where f = u|f | is the polar decomposition off ∈ LG— is holomorphic on the strip ∆ = 0 < Re z < 1 and continuous on itsclosure. Since Fj(it) = u|f |iqt/2uj |hj |it is a partial unitary

supt∈R‖Fj(it)‖LG ≤ 1.

On the other hand, by (5.2) and (5.3) we get for all t ∈ R∥∥Fj(1 + it)∥∥L2(G)

=∥∥|f | q2 hj∥∥L2(G)

≤∥∥(|f | q2 − a)hj∥∥L2(G)

+ ‖ahj‖L2(G) ≤εq/2

2+ ‖a′‖

L2(Gdisc)

≤ εq/2

2+∥∥a′ − |f ′| q2 ∥∥

L2(Gdisc)+∥∥|f ′| q2 ∥∥

L2(Gdisc)≤ εq/2 + ‖f ′‖q/2

Lq(Gdisc).


Therefore, the three-lines lemma implies that for any j ≥ j0

‖Fj(2/q)‖Lq(G) = ‖fuj |hj |2q ‖Lq(G)(5.4)

≤(εq/2 + ‖f ′‖q/2

Lq(Gdisc)

) 2q ≤ ε+ ‖f ′‖

Lq(Gdisc),

which proves Claim A’ i). To prove Claim A’ ii) we proceed exactly as in the proofof Theorem A, but using our version of Claim A’ i). For a fixed trigonometricpolynomial f ′ in LGdisc frequency supported by F, let k′ = k′(f ′, ε, q) be anothertrigonometric polynomial in LGdisc (frequency supported by M ⊂ G finite) andsatisfying ‖k′‖

Lq∗ (Gdisc)= 1 with

‖f ′‖Lq(Gdisc)

≤ ‖k′f ′∗‖L2(Gdisc)

+ε

2,

where 1/q + 1/q∗ = 1/2. We may choose j0 = j0(f ′, ε, q) such that for any j ≥ j0

i) ‖k′f ′∗‖L2(Gdisc)

= ‖k′f ′∗hj‖L2(G),

ii)∥∥kuj |hj |2/q∗∥∥Lq∗ (G)

≤ 1 + ε,

iii)∑g∈F

µ(g−1Vjg4Vj

)µ(Vj)

≤ ε2

‖k‖22‖f‖22|K|.

Namely, the first property follows from (5.1), the second one from (5.4) and thethird one from the SAIN condition. By the same argument as in the proof ofTheorem A, we obtain that for any j ≥ j0

‖f ′‖Lq(Gdisc)

≤ ε+ (1 + ε)‖fuj |hj |2/q‖Lq(G).

Letting ε→ 0+, this implies Claim A’ ii) and completes Theorem D ii).

Remark 5.3. According to Remark 3.4, we know that the Heisenberg group Hn

and the upper triangular matrix groups H(K, n) are ADS. Moreover, since theyare nilpotent the same happens for their discretized forms, which implies in turnthat the discretized forms are amenable. In summary, if G denotes any of thesegroups, it satisfies the two-sided compactification result in Theorem D i) and ii) forbounded continuous symbols∥∥Tm : Lp(G)→ Lp(G)

∥∥ =∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥.6. The periodization theorem

We finish our collection of noncommutative de Leeuw’s theorems in the Banachspace setting for unimodular groups with the periodization theorem, nonunimodulargroups and statements in the operator space setting will be considered below. Inthis section we consider a locally compact, unimodular, second countable groupG; a normal closed subgroup H of G; a bounded symbol mq : G/H → C andits H-periodization mπ : G → C given by mπ(g) = mq(gH). As mentioned inthe Introduction, the abelian case has been solved by Saeki [48] but we cannot gofurther in the line of Theorem D iii). More precisely, in general

Tmq : Lp(G/H)→ Lp(G/H) ; Tmπ : Lp(G)→ Lp(G).


Indeed, consider for instance the infinite permutation group H = S∞ and constructthe cartesian product G = T×S∞, so that G/H ' T. By [45, Proposition 8.1.3], for1 < p 6= 2 <∞ we can find a bounded mq : T→ C giving rise to a Fourier multiplierwhich is bounded in `p(Z) but not completely bounded. Then, its H-periodizationmπ = mq ⊗ id cannot define a bounded Fourier multiplier on

Lp(G) = `p(Z;Lp(R)),

where R = LS∞ denotes the hyperfinite II1 factor. Hence, Theorem D iii) fails forthis pair (G,H). In fact, since Pisier’s result on the existence of bounded/not cbmultipliers has been extended to any infinite LCA groups [1, 25], with that processwe can construct a large class of counter-examples by taking any group of the formG = K×H with K an infinite LCA group and H a group satisfying that LH containsarbitrarily large matrix algebras Mn. This suggests that there is not so much to doin this direction outside the class of abelian groups. The result in Theorem D iii)was already proved by Saeki [48]. Hence, we now focus on the reverse implicationgiven in Theorem D iv) for G nonabelian and H compact.

Proof of Theorem D iv). Assume H is compact and let µH denote the normalizedHaar measure on H. By duality it is enough to consider the case p ≥ 2. By Lemma2.1, we may see LH as a von Neumann subalgebra of LG and identify λG(h) andλH(h) for any h ∈ H. Consider the operator

Π =

∫H

λ(h) dµH(h) ∈ LH ⊂ LG.

Since H is a normal, compact (unimodular) subgroup of G, we deduce that Πis a central, H-invariant projection of LG onto the functions of L2(G) which areconstant on H-cosets, denoted by

H = ΠL2(G) =ξ ∈ L2(G) : ξ(g) = ξ(g′) when gH = g′H

.

The map π : G →M := (LG)Π given by π(g) = λ(g)Π defines a ∗-representationof G over the Hilbert space H. Moreover, π is invariant on cosets, hence this yieldsa ∗-representation of the quotient G/H still denoted by π : G/H → M. Observethat π(gH) = vλG/H(gH)v∗, where the unitary v : L2(G/H) → H is the naturalidentification. Hence π can be extended to a normal map π : L(G/H) → M bysetting π(f) = vfv∗. Since this map is isometric and surjective at the L∞ and L2

levels, this yields by interpolation an isometric map

π : Lp(G/H)→ Lp(M) = Lp(G)Π

for any 2 ≤ p ≤ ∞. On the other hand, π intertwines the Fourier multipliers sothat π Tmq = Tmπ π. Indeed, let f ∈ λ(Cc(G/H)). Since the G-invariant measureon left cosets [18, Theorem 2.49] coincides with the Haar measure on the quotientgroup G/H when H is normal we get

π Tmq (f) =

∫G/H

mq(gH)f(gH)λ(g)Π dµG/H(gH)

=

∫G/H

mq(gH)f(gH)(∫

H

λ(gh) dµH(h))dµG/H(gH)

=

∫G

mπ(g)f(gH)λ(g) dµG(g) Π = Tmπ π(f).


Using this property, we conclude with the estimate

‖Tmqf‖Lp(G/H)= ‖π Tmq (f)‖Lp(M) = ‖Tmπ π(f)‖Lp(G)Π

≤∥∥Tmπ : Lp(G)→ Lp(G)

∥∥‖π(f)‖Lp(G)Π

=∥∥Tmπ : Lp(G)→ Lp(G)

∥∥‖f‖Lp(G/H)

for f ∈ Lp(G/H). This completes the proof of Theorem D iv).

7. Nonunimodular groups

This section is devoted to extend our results to nonunimodular groups. Again themain focus will be on restriction since compactification and periodization admit lessgeneralizations, see Remark 7.13. When G is nonunimodular, the modular function∆G is not trivial and the Plancherel weight —defined in Section 2 and denoted byϕ in this section— is not a trace. This forces to introduce noncommutative Lpspaces associated with arbitrary von Neumann algebras. We will in fact considertwo different such Lp spaces, the Haagerup and the Connes-Hilsum ones [27, 51]which turn to be isomorphic as we explain below. Recall that the proof of TheoremA in the unimodular case is based on crucial results derived from Theorem B.Thus we will need to extend these results to arbitrary von Neumann algebras byusing Haagerup’s reduction method. After that, we will derive Theorem A fornonunimodular groups and give some examples.

7.1. Haagerup’s reduction for weights. We start by recalling the reductionmethod from [24] adapted to a von Neumann algebra M ⊂ B(H) equipped witha fixed normal semifinite faithful (nsf) weight ϕ. Note that the constructions in[24] are carried out with respect to a normal faithful state ϕ instead of a weight.This is not sufficient for our purposes. The weight case is treated in an unpublishedextended version of [24] by Xu. For the sake of completeness, we will indicate belowthe technical modifications of the arguments in [24] to obtain the analogous resultsfor weights instead of states. In this paragraph, we consider the so-called HaagerupLp-spaces defined in [51], see also [24] for a standard introduction of the conceptsinvolved. Since they are only used in this auxiliary technical subsection and thenext one, we will not detail the construction but refer to the above mentionedworks. Let σϕ be the modular automorphism group of ϕ and denote

nϕ = x ∈M : ϕ(x∗x) <∞ and mϕ = n∗ϕnϕ = spany∗x : x, y ∈ nϕ.

In this subsection we fix G = ∪n≥12−nZ with the discrete topology and considerthe crossed product R = M oσϕ G. Recall that R is the von Neumann algebraacting on L2(G,H) generated by the operators(

λ(t)ξ)(s) = ξ(s− t) and

(π(x)ξ

)(s) = σϕ−s(x)ξ(s)

for s, t ∈ G, x ∈M and ξ ∈ L2(G,H). We define the unitary operator(w(γ)ξ

)(s) = γ(s)ξ(s)

for (s, γ, ξ) ∈ G× G×L2(G,H) and αγ(z) = w(γ)zw(γ)∗ for z ∈ R. Then π(M) isthe fixed point algebra for α and the conditional expectation E : R →M is given


by E(x) =∫

Gαγ(x)dγ. The dual weight ϕ on R is defined as ϕ = ϕ π−1 E . Let

Rϕ be the centralizer of ϕ in R and denote by Z(Rϕ) its center. Consider

bn = −iLog(λ(2−n)) and an = 2nbn,

with Log the principal branch of the logarithm, so that 0 ≤ Im(Log(z)) < 2π. Thenbn ∈ Z(Rϕ) and ϕn( · ) = ϕ(e−an · ) formally defines a nsf weight. More precisely,ϕn has Connes cocycle derivative (Dϕn/Dϕ)s = e−isan for s ∈ R.

Theorem 7.1. Let Rn be the centralizer of ϕn in R. The sequence (Rn)n≥1 formsan increasing sequence of von Neumann subalgebras of R. Moreover, the followingproperties hold :

i) Rn is semifinite for each n ≥ 1 with trace ϕn.

ii) There exist conditional expectations En : R → Rn such that

ϕ En = ϕ and En σϕs = σϕs En for all s ∈ R.

iii) En(x)→ x σ-strongly for x ∈ nϕ and⋃n≥1Rn is σ-strongly dense in R.

Proof. The proof is a mutatis mutandis copy of the arguments in [24, Section 2].We indicate the main adaptations. Observe that [24, Lemma 2.2] does not remainvalid. This lemma is applied only in two places, where the arguments need to beadapted. Firstly, it is needed to prove the uniqueness of bn in [24, Lemma 2.3], butthis does not play a role in the subsequent proofs. Secondly it is used in the proofof [24, Lemma 2.6]. However, we claim that the following fact still holds true: forevery x ∈ nϕ and every ε > 0 there exists a trigonometric polynomial P on T with

(7.1)∥∥[bn − P (λ(2−n)), x

]∥∥ϕ≤ ε for all n ∈ N,

where [x, y] = xy − yx denotes the commutator of two operators x and y and‖y‖2ϕ = ϕ(y∗y) for any y ∈ R. This fact is what is actually needed. Let us nowprove it. If x ∈ nϕ, then∥∥(bn − P (λ(2−n))

)x∥∥2

ϕ= ϕ

(x∗|bn − P (λ(2−n))|2x

).

Now ϕ(x∗ · x) is a normal functional on R and hence it restricts to a normalfunctional ω on the von Neumann subalgebra generated by λ(2−n), which equalsL∞(T). So ω corresponds to integration against a function in L1(T). Recallingthat bn = −iLog(λ(2−n)) we see that we may choose P such that for every n wehave ω(|bn − P (λ(2−n))|2) < ε. On the other hand, we first consider

x ∈ Tϕ :=x ∈ R : x is analytic for σϕ and σϕz (x) ∈ nϕ ∩ n∗ϕ ∀ z ∈ C

.

In that case, from Tomita-Takesaki theory we have∥∥x(bn − P (λ(2−n)))∥∥2

ϕ= ϕ

(x|bn − P (λ(2−n))|2σϕ−i(x

∗)),

and as above we may find P such that for every n this expression becomes smallerthan ε. This proves our claim (7.1) in case x ∈ Tϕ. For a general operator x ∈ nϕthe claim follows by taking a net (xj)j∈J in Tϕ such that ‖xj − x‖ϕ → 0 (see forinstance [50]) and using that ‖bn‖ ≤ 2π.

Let us now return to the constructions of [24, Section 2]. The statements andproofs of [24, Lemmas 2.3, 2.4, 2.5] remain unchanged except that bn might not beunique, which is not relevant for the proof. Note in particular that the restriction


of ϕn to its centralizer is semifinite. Then Lemma 2.6 remains true provided x ∈ nϕinstead of general x ∈ R and also Lemma 2.7 remains valid for x ∈ nϕ. Indeed, asin the proof of Lemma 2.7, this follows from Lemma 2.6 in case x ∈ nϕ (and also inthe weight case one invokes Lemma 2.5 to derive strong convergence, which impliesσ-strong convergence for a bounded net). This completes the proof.

Let Lp(M), Lp(R) and Lp(Rn) be the Haagerup Lp-spaces constructed from theweights ϕ, ϕ, and ϕ restricted to Rn respectively, see [51] or [24, Section 1.2]. Themodular automorphism group σϕ restricted toM' π(M) equals σϕ. By Theorem7.1, the restriction of ϕ to Rn is semifinite. This implies that the crossed productsM oσϕ R and Rn oσϕ R are well-defined subalgebras of R oσϕ R. Let D be thegenerator of the left regular representation in each of these crossed products, thenD is the usual density operator in the Haagerup Lp-space Lp(R). Recall that wehave two ϕ-preserving conditional expectations E : R → M and En : R → Rn.For 1 ≤ p < ∞, by Remark 5.6 and Example 5.8 of [24] we obtain contractiveprojections

Ep : Lp(R)→ Lp(M) and Epn : Lp(R)→ Lp(Rn)

given by Ep(D12pxD

12p ) = D

12p E(x)D

12p for any x ∈ mϕ, and similarly for En. More

generally, for any p ≤ r, s ≤ ∞ such that 1r + 1

s = 1p we have Ep(D 1

r xD1s ) =

D1r E(x)D

1s for x ∈ mϕ, see [24, Proposition 5.5] for the proof in the state case.

Remark 7.2. The notation D1r xD

1s for x ∈ mϕ used in [24] and which we keep

using in the sequel is formal. If x can be decomposed as a finite sum x =∑j y∗j zj

with yj , zj ∈ nϕ, then the notation D1r xD

1s stands for

∑j D

1r y∗j · [zjD

1s ], which is

a well-defined element of Lp(R) by [52, Theorem 26] and Holder’s inequality. Here[ · ] denotes the closure of a preclosed operator. Arguing as in [20] one can derivethat this expression does not depend on the decomposition of x.

Lemma 7.3. Given 1 ≤ p <∞ and x ∈ Lp(R) we have

limn→∞

‖Epn(x)− x‖p = 0.

Proof. We first assume that 1 ≤ p ≤ 2. Let x ∈ R and x′, x′′ ∈ Rm for somem ≥ 1. Assume moreover that x, x′′ ∈ nϕ, x

′ ∈ n∗ϕ and n ≥ m. Let r be such that1p −

12 = 1

r . Since En(x′) = x′ and En(x′′) = x′′, using the convention of Remark

7.2, Holder’s inequality and [52, Theorem 23] imply∥∥Epn(D1r x′xx′′D

12 )−D 1

r x′xx′′D12

∥∥p

=∥∥D 1

r x′En(x)x′′D12 −D 1

r x′xx′′D12

∥∥p

≤ ‖D 1r x′‖r

∥∥(En(x)− x)x′′D12

∥∥2

= ‖D 1r x′‖r

∥∥(En(x)− x)Λ(x′′)∥∥

2→ 0,

since En(x) → x strongly by Theorem 7.1. Here Λ denotes the canonical injectionof nϕ into its Hilbert space completion. We claim that the linear span of elements

D1/rx′xx′′D1/2 with x, x′, x′′ as above is dense in Lp(M). Then the result willfollow for any operator x ∈ Lp(R) for 1 ≤ p ≤ 2 by contractivity of Epn. By [52,

Theorem 26] the linear span of D1/ry∗ = [yD1/r]∗ with y ∈ nϕ is dense in Lr(M)

for 2 ≤ r <∞. Then the Holder inequality gives that spanD1/rxD1/2 : x ∈ mϕ


is dense in Lp(M). Let (x′j)j∈J , (x′′j )j∈J be nets in Rn of elements that are analytic

for σϕ and such that

σϕz (x′j), σϕz (x′′j ) ∈ nϕ ∩ n∗ϕ

for every z ∈ C. Assume that σϕ−i/r(x′j) → 1 and σϕi/2(x′′j ) → 1 strongly. Then

using [20, Lemma 2.5] and [34, Lemma 2.3] we get

D1r x′jxx

′′jD

12 = σϕ−i/r(x

′j) ·D

1r xD

12 · σϕi/2(x′′j )→ D

1r xD

12 ,

in the norm of Lp(M). Here, the domains of the operators in the first equalityare equal by an argument similar to the one we will use to prove Lemma 7.6.Thisconcludes our claim, and hence the Lemma for 1 ≤ p ≤ 2. We now consider thecase p ≥ 2. Suppose that we have proved the Lemma for p/2. Take x ∈ R suchthat x ∈ nϕ. By Holder’s inequality∥∥(En(x)− x)D

1p

∥∥2

p=

∥∥(En(x)− x) ·D2p (En(x)− x)∗

∥∥p/2

≤∥∥En(x)− x‖∞‖(En(x)− x)D

2p

∥∥p/2

=∥∥En(x)− x‖∞‖Epn(xD

2p )− xD

2p

∥∥p/2,

which goes to 0 as n tends to ∞. Therefore, the result for a general operatorx ∈ Lp(R) follows by density [52, Theorem 26], recalling that Epn is contractive.

7.2. Almost multiplicative maps on arbitrary von Neumann algebras. Wenow apply the reduction method detailed above to the results of Section 1 needed toprove Theorem A in the nonunimodular setting. LetM be a von Neumann algebrawith a nsf weight ϕ and T :M→M be a positive map such that ϕT ≤ ϕ. Given1 ≤ p <∞ and according to [24, Remark 5.6], the map T induces a bounded mapTp on the Haagerup Lp-space Lp(M) determined by

Tp(D12pϕ xD

12pϕ ) = D

12pϕ T (x)D

12pϕ

for x ∈ mϕ, where Dϕ denotes the density operator of ϕ. With that notation, wecan state and prove the following analogues of Corollary 1.3 and Corollary 1.4 forarbitrary von Neumann algebras.

Corollary 7.4. Let M be a von Neumann algebra equipped with a nsf weight ϕand let T : M → M be a subunital completely positive map with ϕ T ≤ ϕ andT σϕs = σϕs T for every s ∈ R. Then there exists a universal constant C > 0 suchthat the following inequality holds for any x ∈ L+

2 (M) and any 0 < θ ≤ 1∥∥T 2θ(xθ)− xθ

∥∥2θ

≤ C∥∥T2(x)− x

∥∥ θ22‖x‖

θ22 .

Proof. We use the notations of Section 7.1. By [24, Section 4], we know that themap T admits a subunital completely positive normal extension, which is given by

T : R 3 π(x)λ(s) 7→ π(T (x))λ(s) ∈ R

for any (s, x) ∈ G × M. Note that LG is in the multiplicative domain of T .Moreover, we also have

ϕ T ≤ ϕ and σϕt T = T σϕt .


Recall that σϕnt and En are defined in [24] respectively by

σϕns (x) = e−isanσϕs (x)eisan and En(x) = 2n∫ 2−n

0

σϕns (x)ds

for any (x, s) ∈ R × R. Note that these expressions were used in [24] for states,although the same construction is valid for weights and the resulting conditionalexpectations commute with the action of the modular automorphism group. Since

eisan ∈ LG, we deduce that T commutes with En. Hence, we may consider itsrestriction to Rn and deduce that we still have that

ϕn T ≤ ϕn.

By Theorem 7.1 i) (Rn, ϕn) is semifinite and we may extend T to a contractivemap on the tracial Lp-space Lp(Rn, ϕn). This extension does not depend on p. Onthe other hand, for 1 ≤ p <∞ the map given by

Tp(D1/2pϕ xD

1/2pϕ ) = D

1/2pϕ T (x)D

1/2pϕ for x ∈ mϕ

extends to a bounded map Tp : Lp(R, ϕ) → Lp(R, ϕ) by [24, Remark 5.6]. Since

E T = T E , where E : R →M is the ϕ-preserving conditional expectation, the

restriction of Tp to Lp(M) equals Tp. Moreover, it commutes with Epn and we mayconsider the restriction

Tp : Lp(Rn, ϕ)→ Lp(Rn, ϕ).

As it is proved in [51], we have Lp(Rn, ϕ) ' Lp(Rn, ϕn) isometrically, and the iso-

morphism preserves positive elements. The two restriction maps T and Tp are com-patible with respect to that isomorphism. Namely, let κp : Lp(Rn, ϕ)→ Lp(Rn, ϕn)

be the isometric isomorphism given by κp(D1/2pϕ xD

1/2pϕ ) = e

an2p xe

an2p for any x ∈ mϕ,

then

κp Tp = T κp on Lp(Rn, ϕ)

since an lies in the multiplicative domain of T . Fix x ∈ L+2 (M), then by Lemma

7.3 iii) and the fact that Tp commutes with Epn we can write∥∥T 2θ(xθ)− xθ

∥∥L 2θ

(M)=

∥∥T 2θ(xθ)− xθ

∥∥L 2θ

(R)

= limn→∞

∥∥E 2θn T 2

θ(xθ)− E

2θn (xθ)

∥∥L 2θ

(Rn,ϕ)

= limn→∞

∥∥T 2θ

(E

2θn (xθ)

)− E

2θn (xθ)

∥∥L 2θ

(Rn,ϕ)

= limn→∞

∥∥∥T(κ 2θ

(E

2θn (xθ)

))− κ 2

θ

(E

2θn (xθ)

)∥∥∥L 2θ

(Rn,ϕn).

By Corollary 1.3 in (Rn, ϕn) applied to the map T and to κ 2θ

(E

2θn (xθ)

) 1θ ∈ L+

2 (Rn, ϕn)∥∥T 2θ(xθ)− xθ

∥∥2θ

≤ C limn→∞

∥∥∥T(κ 2θ

(E

2θn (xθ)

) 1θ

)− κ 2

θ

(E

2θn (xθ)

) 1θ

∥∥∥ θ2L2(Rn,ϕn)

∥∥∥κ 2θ

(E

2θn (xθ)

) 1θ

∥∥∥ θ2L2(Rn,ϕn)

= C limn→∞

∥∥T2

(E

2θn (xθ)

1θ

)− E

2θn (xθ)

1θ

∥∥ θ2L2(Rn,ϕ)

∥∥E 2θn (xθ)

1θ

∥∥ θ2L2(Rn,ϕ)

.


We finally claim that

limn→∞

∥∥E 2θn (xθ)

1θ − x

∥∥2,

which yields the result since ‖T2(x) − x‖2 = ‖T2(x) − x‖2. This claim followsfrom Lemma 7.3 iii) and the fact that for any operators x, y ∈ L2(R) such that‖y‖2 ≤ ‖x‖2 and any parameter 0 < θ ≤ 1 we have

‖x− y‖2 ≤ k∥∥xθ − yθ∥∥ 1

θk2θ

‖x‖1−1k

2

for any integer k ≥ 1 satisfying 1k ≤ θ. Indeed, we first observe that for 1 ≤ p ≤ ∞

and k ∈ Z+ ∥∥xk − yk∥∥p≤ k‖x− y‖pk‖x‖k−1

pk for x, y ∈ L+pk(R)

with ‖y‖pk ≤ ‖x‖pk. This easily follows from Holder inequality and the identity

xk − yk =

k−1∑j=0

xk−j−1(x− y)yj .

Then we get

‖x− y‖2 ≤ k‖x1k − y 1

k ‖2k‖x‖(k−1)/k2 ≤ k‖xθ − yθ‖1/θk2/θ ‖x‖

(k−1)/k2 .

The last inequality follows from the Powers-Størmer inequality Lemma 1.2. Notethat we have not justified the validity of such inequality for type III algebras. It ishowever a simple exercise to deduce it from [35, Proposition 7 and Lemma B].

Corollary 7.5. Let M be a von Neumann algebra equipped with a nsf weight ϕand let T : M → M be a subunital completely positive map with ϕ T ≤ ϕ andT σϕs = σϕs T for every s ∈ R. Then there exists a universal constant C > 0such that the following inequality holds for any self-adjoint y ∈ L2(M) with polardecomposition y = u|y| and any 0 < θ ≤ 1∥∥T 2

θ(u|y|θ)− u|y|θ

∥∥2θ

≤ C∥∥T2(y)− y

∥∥ θ42‖y‖

3θ4

2 .

Proof. The proof is similar to the one of Corollary 1.4, details are omitted.

7.3. Connes-Hilsum Lp spaces. In this subsection we recall the constructionfor group von Neumann algebras of Connes-Hilsum Lp-spaces [27], since we shalluse them in the proof of Theorem A. This construction will also be needed in thenext section, in order to apply the transference results from [7] in the category ofoperator spaces. Since our proof of Theorem A will rely on the results derivedfrom Theorem B established in Section 7.2 for the Haagerup Lp-spaces, we need tocompare both constructions.

The Connes-Hilsum construction for group algebras. We shall follow thepresentation of [7]. Let G be a locally compact group and let ρ : G→ B(L2(G)) bethe right regular representation

ρ(g)(ξ)(h) = ∆G(g)12 ξ(hg)

for any ξ ∈ L2(G) and g, h ∈ G. Set

ρ(ξ) =

∫G

ξ(g)ρ(g)dµ(g) for any ξ ∈ L2(G).


There exists a nsf weight ϕ′ on the commutant LG′ = ρ(G)′′ given by

ϕ′(f∗f) =

∫G

|ξ(g)|2 dµ(g)

when f = ρ(ξ) for some ξ ∈ L2(G) and ϕ′(f∗f) =∞ for any other f ∈ LG′. For a

nsf weight ω on LG, the partial derivative (dω/dϕ′)12 is the unique closed densely

defined operator, whose domain consists of the left bounded functions in L2(G) andsuch that ∥∥(dω/dϕ′)

12 ξ∥∥2

L2(G)= ω(λ(ξ)λ(ξ)∗) <∞.

For 1 ≤ p < ∞, the Connes-Hilsum noncommutative space Lp(G) = Lp(G, ϕ′)

is then defined as the set of closed densely defined operators f on L2(G) withpolar decomposition f = u|f | such that u ∈ LG and |f |p equals dω/dϕ′ for someω ∈ LG∗. In that case

‖f‖Lp(G) = ω(1)1p = ‖ω‖

1p .

Equipped with this norm, Lp(G) is a Banach space and the Holder inequality holdsby understanding the product of two operators as the closure of their product. Forξ ∈ L1(G) ∩ L2(G), we have the Plancherel formula

[λ(ξ)∆12

G] ∈ L2(G) with∥∥[λ(ξ)∆

12

G]∥∥L2(G)

= ‖ξ‖L2(G).

In fact, such elements are dense in L2(G). Moreover, the set of operators[λ(ξ)∆

1p

G] : ξ ∈ Cc(G)

is dense in Lp(G) for 2 ≤ p < ∞, see [52, Theorem 26]. Connes-Hilsum Lp-spacesare compatible with interpolation, meaning that we may find a compatible structure

so that the family (Lp(G))1≤p≤∞ forms an interpolation scale, further details canbe found in [7]. Let 2 ≤ p ≤ ∞ and consider any symbol m ∈ L∞(G). Given anyξ ∈ Cc(G), we have

[λ(ξ)∆1/pG ], [λ(mξ)∆

1/pG ] ∈ Lp(G).

Then we may consider the associated multiplier

T pm : Lp(G) 3 [λ(ξ)∆1/pG ] 7→ [λ(mξ)∆

1/pG ] ∈ Lp(G),

which is called an Lp-Fourier multiplier if it extends boundedly to Lp(G) (to anormal map if p = ∞). For 1 ≤ p ≤ 2 and a given bounded symbol m, we definethe associated multiplier by

T 1m := (T∞mop

)∗ and T pm = (T p′

mop)∗ where mop(s) = m(s−1).

Relation between Haagerup and Connes-Hilsum spaces. Let us fix somenotation. We let M be a von Neumann algebra equipped with nsf weight ϕ. Letϕ′ be a nsf weight on the commutantM′. We let Lp(M) be the Connes-Hilsum Lpspace constructed from ϕ′. Let d = dϕ/dϕ′ be the spacial derivative. Let x ∈ mϕand write x =

∑i y∗i zi with yi, zi ∈ nϕ (finite sum). Define

jp(x) =∑

id

12p y∗i · [zid

12p ] ∈ Lp(M).

The sum above does not depend on the choice of yi and zi, see [20]. We let Lp(M)obe the Haagerup Lp-space constructed from ϕ with density operator D. Define also

jp,o(x) =∑

iD

12p y∗i · [ziD

12p ] ∈ Lp(M).


Let Sϕ be the set of all x ∈ M such that x is analytic for σϕ and σϕz (x) ∈ mϕ forevery z ∈ C. Recall that if a is a closed unbounded operator and b is a boundedoperator then ab is automatically closed.

Lemma 7.6. For every x ∈ Sϕ we have

jp(x) = d1pσϕi

2p

(x) = [σϕ− i2p

(x)d1p ].

Proof. Let yi, zi ∈ nϕ be such that

x =∑

iy∗i zi.

Using [52, Lemma 22] for the first inclusion and an elementary inclusion

σϕ− i2p

(x)d1p ⊆ d

12pxd

12p ⊆

∑id

12p y∗i · [zid

12p ] ∈ Lp(M).

Hence (σϕ− i

2p

(x)d1p)∗ ⊇∑

i

(d

12p y∗i · [zid

12p ])∗ ∈ Lp(M).

By (the proof of) [27, Theorem 4 (1)] we in fact have an equality(σϕ− i

2p

(x)d1p)∗

=∑

i

(d

12p y∗i · [zid

12p ])∗.

Therefore, taking adjoints yields the equality in the next line

d1pσϕi

2p

(x) ⊇ [σϕ− i2p

(x)d1p ] =

∑id

12p y∗i · [zid

12p ],

whereas the first inclusion follows from [52, Lemma 22]. Because the right handside is in Lp(M) this inclusion is in fact an equality by [27, Theorem 4 (1)].

Lemma 7.7. For every x ∈ Sϕ we have

jp,o(x) = D1pσϕi

2p

(x) = [σϕ− i2p

(x)D1p ].

Proof. The proof is the same as of Lemma 7.6. The only difference being thatevery time that we used [27, Theorem 4 (1)] one uses [51, Proposition I.12].

Proposition 7.8. Let T :M→M be a completely bounded map with ϕ T ≤ ϕthat commutes with σϕ. Let Tp,o : Lp(M)o → Lp(M)o be the extended map tothe Haagerup Lp-space given in [24, Remark 5.6] and which is determined by therelation below for x ∈ Sϕ

Tp,o : jp,o(x) 7→ jp,o(T (x)).

Then, the isometric isomorphism

κp : Lp(M)→ Lp(M)o

defined in [51] intertwines Tp and Tp,o, where

Tp : Lp(M) 3 jp(x) 7→ jp(T (x)) ∈ Lp(M) for x ∈ Sϕ.

Proof. Note that the statement above uses that T preserves the set Sϕ, which isclear from the definition. Let u0 : L2(R,H) → L2(R,H) be the map defined by(u0ξ)(s) = disξ(s) with s ∈ R. Let D0 be such that Dis

0 = λ(s), where

(λ(s)ξ)(t) = ξ(t− s)


is the left regular representation on L2(R). It is proved in [51, Proposition IV.3]that

u0π(x)u∗0 = x⊗ 1,

u0λ(s)u∗0 = dis ⊗ λ(s),

where x ∈M and s ∈ R. Let κp : Lp(M)→ Lp(M)o be the isometric isomorphism

defined in [51] by a 7→ u∗0(a ⊗D1/p0 )u0. Let 1 ≤ p < ∞ and x ∈ Sϕ. Consider the

element d1/px which is in Lp(M) by Lemma 7.6. Then

Tp,o κp(d

1px)

= Tp,o

(u∗0(d

1px⊗D

1p

0

)u0

)= Tp,o

(u∗0(d

1p ⊗D

1p

0

)u0u∗0

(x⊗ 1

)u0

)= Tp,o

(D

1pπ(x)

)= D

1pπ(T (x)

)= u∗0

(d

1p ⊗D

1p

0

)u0u∗0

(T (x)⊗ 1

)u0

= u∗0(d

1pT (x)⊗D

1p

0

)u0 = κp Tp

(d

1px).

Note in particular that at each instance we have an equality of domains and thefourth equality follows from Lemma 7.7 and the definition of T . Similarly, the lastequality follows from Lemma 7.6. Since such elements d1/px are dense in Lp(M)this proves that Tp,o κp = κp Tp. This completes the proof.

Remark 7.9. In particular, Corollary 7.5 is valid for Connes-Hilsum Lp-spaces.

7.4. Nonunimodular restriction theorem. We finish this section by sketchingthe proof of the restriction theorem in the nonunimodular setting, enlightening themain changes. Note that in the nonunimodular case, the Fourier multipliers dependon p. However, for the sake of clarity we just used the notation Tm in the statementof Theorem A given in the Introduction. After the proof, we shall construct somenatural examples illustrating Theorem A which complement what we did in Section4. We shall also give a brief discussion on Theorem D in Remark 7.13.

Proof of Theorem A: Nonunimodular case. The proof follows the samestrategy as in the unimodular case, the main ingredient being that in this case theoperator hj should be defined as

hj =∥∥1Vj∆− 1

4

G

∥∥−1

L2(G)

[λ(1Vj∆

− 14

G )∆12

G

]∈ L2(G).

Note that hj is a self-adjoint operator. Indeed, according to [20, Lemma 2.5] and

the fact that Vj is symmetric (recalling that ξ∗(g) = ∆G(g)−1ξ(g−1)), we obtainthe following identity

h∗j =∥∥1Vj∆− 1

4

G

∥∥−1

L2(G)∆

12

Gλ(1Vj∆− 1

4

G )∗

=∥∥1Vj∆− 1

4

G

∥∥−1

L2(G)∆

12

Gλ(1Vj∆− 3

4

G )

=∥∥1Vj∆− 1

4

G

∥∥−1

L2(G)

[λ(1Vj )∆

− 14

G )∆12

G

]= hj .

Then one defines again Φqj : Lq(Γ) 3 f 7→ fuj |hj |2q ∈ Lq(G), where hj = uj |hj |

is the polar decomposition. The proof proceeds then exactly as in the unimodularcase in Section 4, which relies on two claims. We can check that Claim A and


Claim B can be proved mutatis mutandis, except that we need Corollary 1.4 fornoncommutative Lp-spaces associated with type III algebras. Recall that we appliedthis result to the Fourier multiplier Tζ for some unital, continuous, positive definitefunction ζ ∈ L∞(G). Then Tζ is a unital completely positive, ϕ-preserving map andby [24, Example 5.9] it commutes with the modular automorphism group. Thus,we may apply Corollary 7.5 in conjunction with Remark 7.9.

We now illustrate Theorem A with some example, the main of which will be toshow that we can apply restriction to any ADS amenable subgroup for which themodular function restricts properly. We need a preliminary technical result.

Lemma 7.10. Let G be a locally compact group. Let ε > 0 and ξ, η1, · · · , ηn bepositive functions in L1(G) satisfying

∑n`=1 ‖ξ−η`‖1 < ε and ‖ξ‖1 = 1. Then there

exists t > 0 such thatn∑`=1

∥∥1ξ>t − 1η`>t∥∥L1(G)

< ε‖1ξ>t‖L1(G).

Proof. Given g ∈ G and 1 ≤ ` ≤ n, we have

|ξ(g)| =

∫ ∞0

1ξ(g)>tdt,

|ξ(g)− η`(g)| =

∫ ∞0

∣∣1ξ(g)>t − 1η`(g)>t∣∣ dt.

Hence the hypothesis can be written as follows∫ ∞0

n∑`=1

‖1ξ>t − 1η`>t‖L1(G)dt =

n∑`=1

‖ξ − η`‖L1(G)

< ε‖ξ‖L1(G) = ε

∫ ∞0

‖1ξ>t‖L1(G) dt.

This immediately implies the existence of some t > 0 satisfying the assertion.

Theorem 7.11. We have

G ∈ [SAIN]Γ

for any discrete amenable subgroup Γ satisfying that ∆G|Γ= ∆Γ = 1.

Proof. Fix a finite set F ⊂ Γ. Since Γ is amenable and discrete, we know fromthe Følner condition (see Lemma 2.2) that for any j ≥ 1 there exist a finite subsetUF,j ⊂ Γ such that

|UF,jγ4UF,j ||UF,j |

<1

j|F|for any γ ∈ F.

Let (Zj)j≥1 be a basis of symmetric neighborhoods of e such that µ(Zj) < ∞.Since the sets (UF,j)j≥1 are finite, by continuity of the multiplication on G wecan find a sequence (Wj)j≥1 of symmetric neighborhoods of the identity such that⋃g∈UF,j

g−1Wjg ⊂ Zj . Define for each j ≥ 1

ξj =1

|UF,j |µ(Wj)

∑g∈UF,j

1g−1Wjg.


By unimodularity of Γ (since it is discrete), we have ‖ξj‖L1(G) = 1 and we canprove ∥∥ξj(γ · γ−1)− ξj

∥∥L1(G)

<1

j|F|for any γ ∈ F.

Indeed, using that for any j ≥ 1 we have

ξj(γ · γ−1)− ξj =1

|UF,j |µ(Wj)

( ∑g∈UF,jγ\UF,j

1g−1Wjg −∑

g∈UF,j\UF,jγ

1g−1Wjg

),

and by construction of the Følner sets (UF,j)j≥1 we get∥∥ξj(γ · γ−1)− ξj∥∥L1(G)

≤ |UF,jγ4UF,j ||UF,j |

<1

j|F|.

Here we also used that UF,jγ ∪UF,j ⊂ Γ for any γ ∈ F and the unimodularity of Γ.Hence, by applying Lemma 7.10, for each j ≥ 1 we can find tj > 0 such that theset Vj = ξj > tj satisfies

(7.2)∑γ∈F

µ(γ−1Vjγ4Vj

)µ(Vj)

=∑γ∈F

‖1Vj − 1γ−1Vjγ‖L1(G)

‖1Vj‖L1(G)<

1

j.

It remains to check that (Vj)j≥1 is a basis of symmetric neighborhoods of theidentity. Since Wj is symmetric, we have ξj(g

−1) = ξj(g) for any g ∈ G and Vj isclearly symmetric. On the other hand, note that ‖ξj‖∞ = ξj(e) = µ(Vj)

−1. Thusξj(e) > tj , otherwise we would have 1Vj = 0, which contradicts (7.2). Finally, theinclusions

Vj ⊂ supp(ξj) ⊂⋃

g∈UF,j

g−1Wjg ⊂ Zj

ensure that (Vj)j≥1 is a basis of neighborhoods of the identity.

Corollary 7.12. We have∥∥Tm|H : Lp(H)→ Lp(H)∥∥ ≤ ∥∥Tm : Lp(G)→ Lp(G)

∥∥for any ADS amenable subgroup H satisfying the identity ∆H = ∆G|H

.

Proof. According to Theorem C, we have∥∥Tm|H : Lp(H)→ Lp(H)∥∥ ≤ sup

j≥1


for any family (Γj)j≥1 of discrete subgroups approximating H. By amenability andunimodularity of H, it is easily seen that each Γj satisfies the hypothesis of Theorem7.11, from which the assertion follows. This completes the proof.

Beyond discrete amenable subgroups of unimodular groups, other pairs (G,H)satisfying Corollary 7.12 are given by G unimodular and H belonging to the familiesin Remark 3.4. Corollary 7.12 also admits pairs with G nonunimodular, consider forinstance the affine group G = RnoGLn(R) which is nonunimodular [18]. However∆G restricts to SLn(R) (which is unimodular) trivially and hence also to every ADSsubgroup. In particular ADS subgroups of On(R) will form examples of subgroupsof G that satisfy the criteria of Theorem A.


Remark 7.13. Outside the cb-setting (Section 8), the compactification Theorem Di) and ii) does not have a suitable analogue in the nonunimodular setting (at leastnot from our techniques) since we require that ∆G = ∆Gdisc

≡ 1. Similarly theperiodization Theorem D iii) is meaningless, since we showed that commutativityof G (hence unimodularity) is an essential assumption. However, Theorem D iv)does generalize to nonunimodular groups provided that ∆G|H

= ∆H, the proof isanalogous to the one we gave for unimodular groups.

8. Operator space results

The goal of this section is to study de Leeuw’s theorems for locally compactgroups in the category of operator spaces. More precisely, we are interested inrestriction, compactification and periodization results under the assumption thatour multipliers are not only bounded, but completely bounded when we equip ourLp spaces with their natural operator space structure [45, 46]. Then we aim toshow that the conclusions also give cb-bounded multipliers. It is easily seen thatthis is the case when we keep the hypotheses of Theorems A, C and D. In otherwords, we have for 1 ≤ p ≤ ∞:

• If H ∈ ADS and G ∈ [SAIN]H, we have∥∥Tm|H : Lp(H)→ Lp(H)∥∥

cb≤∥∥Tm : Lp(G)→ Lp(G)

∥∥cb

for bounded continuous symbols m : G→ C provided ∆G|H= ∆H.

• If G ∈ ADS is approximated by (Γj)j≥1∥∥Tm : Lp(G)→ Lp(G)∥∥

cb≤ sup

j≥1


cb

for bounded m : G→ C which are continuous µG–almost everywhere.

• If G is ADS, Gdisc is amenable and m continuous∥∥Tm : Lp(G)→ Lp(G)∥∥

cb=∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥cb.

The ≤ holds for G ∈ ADS, the ≥ for Gdisc amenable and G unimodular.

• If H C G is compact and mπ(g) = mq(gH) is bounded∥∥Tmπ : Lp(G)→ Lp(G)∥∥

cb≥∥∥Tmq : Lp(G/H)→ Lp(G/H)

∥∥cb.

Indeed, except for Theorem D iii) our results remain valid when we apply themto the cartesian product of G with any finite group, since our ADS and SAINassumptions are stable under that operation. This operation allows to generalizeour results to the cb-setting in a trivial way.

Remark 8.1. The upper estimate ≤ in our cb-periodization result can be extendedto any pair (G,H) as long as G is discrete, LG is QWEP and ∆G = ∆H on H.Indeed, the discreteness of G and G/H allows us to apply Fell’s absorption principlein Lemma 2.3 ii) to the strongly continuous representation π : g 7→ λG/H(gH) andthe existence of an invariant measure is then used to factorize the integral over Gas an integral over G/H×H. After rearrangement and Fubini’s theorem (for whichwe use the QWEP property following [30]) one concludes.


Motivated by the transference results from [5, 7, 40] between Fourier and Schurmultipliers, an alternative approach to obtain de Leeuw type theorems is to exploitthat such results are much more elementary for Schur multipliers. Namely, givena bounded symbol m : G → C, recall that the associated Herz-Schur multiplier isformally defined as the linear map

Sm :∑

g1,g2∈G

ag1,g2eg1,g2 7→∑

g1,g2∈G

m(g−11 g2)ag1,g2eg1,g2 .

By the boundedness of m, it is clear that Sm is (completely) bounded on theSchatten class S2(L2(G)). When it maps S2(L2(G))∩Sp(L2(G)) to Sp(L2(G)) andextends to a cb-map on Sp(L2(G)), we say that Sm is a cb-bounded Schur multiplieron Sp(L2(G)). Let us analyze de Leeuw operations for Schur multipliers.

Lemma 8.2. If 1 ≤ p ≤ ∞ and m : G→ C is continuous∥∥Sm : Sp(L2(G))→ Sp(L2(G))∥∥

cb

=∥∥Sm : Sp(`2(Gdisc))→ Sp(`2(Gdisc))

∥∥cb.

Moreover, let H be a closed subgroup of G. Then we additionally have

i) If m : G→ C is continuous∥∥Sm|H : Sp(L2(H))→ Sp(L2(H))∥∥

cb≤∥∥Sm : Sp(L2(G))→ Sp(L2(G))

∥∥cb.

ii) If H C G and mq : G/H→ C is continuous∥∥Smπ : Sp(L2(G))→ Sp(L2(G))∥∥

cb=∥∥Smq : Sp(L2(G/H))→ Sp(L2(G/H))

∥∥cb.

Proof. Lafforgue and de la Salle established in [36, Theorem 1.19] (extending anunpublished result of Haagerup in the L∞-case) that for any locally compact groupG and any continuous symbol m : G → C, the cb-norm of the Schur multiplier isgiven by∥∥Sm : Sp(L2(G))→ Sp(L2(G))

∥∥cb

(8.1)

= supF⊂G

F finite

∥∥Sm|F : Sp(`2(F))→ Sp(`2(F))∥∥

cb.

The first assertion (compactification) and property i) (restriction) follow directlyfrom this. The cb-periodization ii) for Schur multipliers can also be deduced from(8.1) as follows. For a fixed fundamental domain X, we consider the natural mapσ : G/H → X. Then we may identify the group G with the cartesian productG/H×H as in the proof of Lemma 2.1 via the bijective map

Υ : G 3 g 7→ (gH, h(g)) ∈ G/H×H

where g = σ(gH)h(g). For 1 ≤ p ≤ ∞, this gives a map

Υ : Sp(L2(G))→ Sp(L2(G/H)⊗ L2(H))

which is completely isometric on finite subsets. Moreover, this map intertwines theSchur multipliers Υ Smπ Υ−1 = Smq ⊗ idB(L2(H)). Therefore, by (8.1) we canwrite∥∥Smq : Sp(L2(G/H))→ Sp(L2(G/H))

∥∥cb

= supn≥1

sup(F1,F2)∈G/H×H

F1,F2 finite

sup‖A‖Sp≤1

A∈M|F1||F2|n

∥∥Smq ⊗ idM|F2|⊗ idMn(A)

∥∥Sp(`2(F1)⊗`2(F2)⊗`n2 )

.


On the other hand, each term of this supremum satisfies∥∥Smq ⊗ idM|F2|⊗ idMn(A)

∥∥Sp(`2(F1)⊗`2(F2)⊗`n2 )

=∥∥(Υ⊗ idMn)(Smπ ⊗ idMn)(Υ−1 ⊗ idMn)(A)

∥∥Sp(`2(F1)⊗`2(F2)⊗`n2 )

=∥∥(Smπ ⊗ idMn)(A)

∥∥Sp(`2(F)⊗`n2 )

,

where F = Υ−1(F1 × F2) ⊂ G is finite and

A = Υ−1 ⊗ idMn(A) ∈ Sp(`2(F)⊗ `n2 )

is of norm 1. Hence we deduce that∥∥Smq : Sp(L2(G/H))→ Sp(L2(G/H))∥∥

cb=∥∥Smπ : Sp(L2(G))→ Sp(L2(G))

∥∥cb.

Indeed, the left hand side is clearly dominated by the right hand side. The lowerestimate also holds since for any finite subset F ⊂ G, we can find finite subsetsF1 ⊂ G/H and F2 ⊂ H such that

F ⊂ Υ−1(F1 × F2).

Thus, the result follows using that the cb-norm in (8.1) is increasing with F.

This shows that de Leeuw theorems extend in almost full generality to the contextof Schur multipliers, only continuity of the symbols is needed. In particular, wedo not impose any of our former conditions like ADS, SAIN, the compatibilityof modular functions or the amenability of Gdisc. We now want to use certaintransference results to obtain de Leeuw type theorems for Fourier multipliers fromthe results in Lemma 8.2. More precisely, we will use that we have

(8.2)∥∥Tm : Lp(G)→ Lp(G)

∥∥cb

=∥∥Sm : Sp(L2(G))→ Sp(L2(G))

∥∥cb

for 1 ≤ p ≤ ∞ under the following conditions

(i) G is an amenable group,

(ii) m ∈ L∞(G) defines a completely bounded Fourier multiplier on Lp(G).

When p = 1,∞ this was proved by Bozejko and Fendler [5]. Other values of pwere first considered by Neuwirth and Ricard [40], who proved (8.2) for amenablediscrete groups. Caspers and de la Salle [7] then obtained this result for arbitraryamenable groups and 1 < p < ∞. We shall need this identity to transfer Lemma8.2 to Fourier multipliers. Hence the price to avoid our conditions listed at thebeginning of this section is to assume amenability of G.

Remark 8.3. Observe that the transference theorem proved in [7] requires theextra assumption that the symbol m : G → C gives rise to a completely boundedFourier multiplier on LG. The set of such symbols is denoted by Mcb(G). Byapproximation we may extend the identity (8.2) to any bounded symbol m : G→ Csatisfying the above condition (ii) whenever G is amenable. Indeed, consider asymbol m ∈ L∞(G) verifying (ii). Notice that when G is amenable there is acontinuous contractive approximate unit (mi)i≥1 in the Fourier algebra A(G) withcompact support. Take also (χj)j≥1 a contractive approximate unit in L1(G) thatalso belongs to L2(G). Define

mi,j = χj ∗ (mim) ∈ L∞(G).


Clearly mi,j ∈ A(G) and hence lies in Mcb(G). On the other hand, one can checkthat ∥∥Tm : Lp(G)→ Lp(G)

∥∥cb

= limi,j

∥∥Tmi,j : Lp(G)→ Lp(G)∥∥

cb,∥∥Sm : Sp(L2(G))→ Sp(L2(G))

∥∥cb

= limi,j

∥∥Smi,j : Sp(L2(G))→ Sp(L2(G))∥∥

cb.

Indeed, the lower estimates easily follows from standard properties of Fourier andSchur multipliers, and we may deduce the upper estimates from the fact that

Tmi,j → Tm (resp. Smi,j → Sm) pointwise in the weak-topology of Lp(G) (resp.Sp(L2(G))). Using Caspers and de la Salle’s result for the symbols mi,j in Mcb(G),this allows us to conclude that (8.2) holds true for the symbol m.

Theorem 8.4. Let 1 ≤ p ≤ ∞ and G amenable:

i) If m : G→ C is bounded and continuous and H is a closed subgroup of G∥∥Tm|H : Lp(H)→ Lp(H)∥∥

cb≤∥∥Tm : Lp(G)→ Lp(G)

∥∥cb.

ii) If m : G→ C is bounded and continuous and Gdisc is amenable, we have∥∥Tm : Lp(G)→ Lp(G)∥∥

cb=∥∥Tm : Lp(Gdisc)→ Lp(Gdisc)

∥∥cb.

iii) If mq : G/H → C is bounded and continuous and H is a normal closedsubgroup of G∥∥Tmπ : Lp(G)→ Lp(G)

∥∥cb

=∥∥Tmq : Lp(G/H)→ Lp(G/H)

∥∥cb.

Proof. It follows from Lemma 8.2, the transference theorem (8.2) from [7] andRemark 8.3.

Remark 8.5. Recall that the lattice approximation Theorem C only works inthe unimodular setting (since we need to assume G ∈ ADS), hence applying thetransference in that case would not improve the cb-result obtained directly fromTheorem C. In fact, applying the transference theorem from [7] and Remark 8.3in conjunction with Theorem 8.4 i) to that result, we deduce the analog for Schurmultipliers. Namely, for any group G ∈ ADS approximated by (Γj)j≥1, 1 ≤ p ≤ ∞and any bounded a.e. continuous symbol m : G→ C, we have∥∥Sm : Sp(L2(G))→ Sp(L2(G))

∥∥cb

= supj≥1

∥∥Sm|Γj : Sp(`2(Γj))→ Sp(`2(Γj))∥∥

cb.

A. Idempotent multipliers in R

Idempotent Fourier multipliers are those whose symbols are the characteristicfunctions of a measurable set Σ. Intervals in R or polyhedrons in Rn are examplesof idempotent symbols which yield Lp-bounded Fourier multipliers (1 < p <∞) asa consequence of the boundedness of the Hilbert transform. When n > 1, we knowfrom the work of Fefferman [16] a fundamental restriction for Lp-boundedness ofidempotent Fourier multipliers over (say) convex sets Σ with boundary ∂Σ. Namely,let

∂Σ⊥ =v ∈ Sn−1

∣∣ v ⊥ ∂Σ.

Then, given Π ⊂ Rn any 2-dimensional vector space, Ω = ∂Σ⊥ ∩ Π can not admitKakeya sets of directions in the sense of [16] or [21, Lemma 10.1.1] when Σ leads


to an Lp-bounded idempotent multiplier. To be more precise we need a bit ofterminology. Given a rectangle R in R2, denote by R′ one of the two translationsof R which are adjacent to R along its shortest side. After a careful reading of theargument in [16, 21], we could say that a subset Ω of the unit circle in R2 admitsKakeya sets of directions when for every N ≥ 1 there exists a finite collection ofpairwise disjoint rectangles RΩ(N) with longest side pointing in a direction of Ωand a family R′Ω(N) formed by rectangles R′ adjacent to the members of RΩ(N)along their shortest side and such that∣∣∣ ⋃

R∈RΩ(N)

R∣∣∣ ≥ N

∣∣∣ ⋃R′∈R′Ω(N)

R′∣∣∣.

The above symbol | | refers to the Lebesgue measure. This notion is closely relatedto Bateman’s notion of Kakeya sets of directions [2]. Fefferman’s theorem impliesthat ∂Σ must have vanishing curvature for Lp-boundedness, as for polyhedrons.Other regions with flat boundary —polytopes with infinitely many faces— may ormay not admit Kakeya sets of directions. This is very connected to the boundednessof directional maximal operators [2, 43] but we shall not analyze these subtletieshere. Apart from the geometric aspect of Σ, one may consider which topologicalstructures of Σ yield Lp-boundedness. In dimension 1, Lebedev and Olevskii [37]showed that Σ must be open up to a set of zero measure, see also Mockenhauptand Ricker [39] for Lp-bounded idempotents which are not Lq-bounded.

Our aim in this Appendix is motivated by a problem left open in [32]. Theauthors provided there a noncommutative Hormander-Mihlin multiplier theoremusing group cocycles in discrete groups as substitutes of more standard geometrictools for Lie groups. This gave rise to some exotic Euclidean multipliers which areLp-bounded in Rn. Consider the cocycle b : R→ R4 given by

s 7→ b(s) =(

cos(2πs)− 1, sin(2πs), cos(2πβs)− 1, sin(2πβs))

associated with the action α : R y R4 ' C2

αs(x1, x2, x3, x4) ' αs(z1, z2) =(e2πisz1, e

2πiβsz2

).

Then, any symbol of the form m(s) = m(b(s)) satisfying that

|∂βs m(s)| . |s|−|β| for s ∈ R4 \ 0 and 0 ≤ |β| ≤ 3

defines an Lp-bounded Fourier multiplier in R for 1 < p <∞. Take for instance ma Hormander-Mihlin smoothing of the characteristic function of an open set Σ inR4 intersecting the range of b. If β ∈ R \Q, the cocycle b has a dense orbit and moscillates from 0 to 1 infinitely often with no periodic pattern. A moment of thoughtshows that the Lp-boundedness of such a multiplier follows from the combination ofde Leeuw’s restriction and periodization theorems, but this cocycle formulation ledJunge, Mei and Parcet to pose a similar problem in [32] when the lifted multiplierm is not smooth anymore. More precisely, let m be the characteristic function ofcertain set Σ which yields an Lp-bounded multiplier in R4 and intersects the rangeof the cocycle b. Is m = m b an Lp-bounded idempotent multiplier on R?

In order to answer the question above, let us formulate the problem in a moretransparent way. The image of the cocycle b is an helix in a two-dimensional toruswhich up to a translation we may identify with T2 ' [0, 1]2. Moreover, under thisidentification, the helix corresponds to the straight line γ in R2 passing through


the origin with slope β. Let us consider the set Ω which results of the intersectionbetween Σ and the two-dimensional torus where b takes values. We shall identifythis set with the corresponding set in [0, 1]2, still denoted by Ω. According to theresults in [16], we know that Σ must have flat boundary. Assume for simplicitythat Σ is a simple object like a semispace or a convex polyhedron —finite unionsand certain infinite unions of this kind of sets also define Lp-bounded idempotentmultipliers— so that Ω is a closed simply connected set. In summary, given a simplyconnected set Ω in [0, 1]2 and certain slope β, we may consider the idempotentFourier multiplier associated with the symbol determined by Figure I below andgiven by

MΩ,β(s) = 1Ω

((s, βs) + Z2

)for s ∈ R.

Ω

slope(γ)=β

γ

6

-

Figure IThe idempotent symbol MΩ,β

MΩ,β = 1 when γ intersects Ω + Z2 and 0 otherwise

Our problem is to decide for which pairs (Ω, β) we get Lp-bounded idempotentmultipliers on R. There are two cases for which the answer is simple. If the slopeβ ∈ Q, the helix is periodic and so is MΩ,β . Therefore, the Lp-boundedness followsby the boundedness of the Hilbert transform for 1 < p <∞ (finitely many times) inconjunction with de Leeuw’s periodization in R. On the other hand, we also obtainLp-boundedness when Ω is a polyhedron (finitely many faces) since we know itscharacteristic function defines an Lp-bounded idempotent multiplier in R2 (finitelymany directional Hilbert transforms). Namely, its Z2-periodization in Lp(R2) andits restriction to γ in Lp(R) are still bounded by de Leeuw’s periodization andrestriction theorems. In particular, the interesting case arises for sets Ω admittingKakeya sets of directions —either having smooth boundary with non-zero curvatureas in Figure I or with infinitely many flat faces admitting Kakeya sets— and slopeβ ∈ R \ Q. We will answer this problem in the negative by combining de Leeuw’srestriction, lattice approximation and Fefferman’s construction.

Theorem A.1. Assume that

β ∈ R \Q and Ω ⊂ [0, 1]2 admits Kakeya sets of directions.

Then MΩ,β does not give rise to a bounded multiplier in Lp(R) for 1 < p 6= 2 <∞.


Proof. Assume there exists 1 < p0 6= 2 <∞ such that

(A.1)∥∥TMΩ,β

: Lp0(R)→ Lp0

(R)∥∥ ≤ C0 < ∞.

Let MΩ,β(L) = MΩ,β 1[0,L] be the L-truncation of our multiplier. If HTp0denotes

the norm of the Hilbert transform on Lp0(R), it is clear that we have the following

bound

supL>0

∥∥TMΩ,β(L) : Lp0(R)→ Lp0

(R)∥∥ ≤ 2 C0 HTp0

.

On the other hand, we may consider the polygon ΠΩ(L, β) determined by thecrossing points of γ + Z2 with Ω in [0,L]. More precisely, let us set

ΠΩ(L, β) = Conv(

Ω ∩

(s, βs) + Z2 : s ∈ [0,L]).

It is illustrated in Figure II and MΩ,β(L) = MΠΩ(L,β),β(L). By the irrationality

of β, the set γ + Z2 is dense in [0, 1]2 and ΠΩ(L, β) converges uniformly to Ω asL→∞. In particular, by constructing finer and finer Kakeya sets of directions, wemay pick L0 large enough so that the following inequality holds

(A.2) infL≥L0

∥∥T1ΠΩ(L,β): Lp0

(R2)→ Lp0(R2)

∥∥ > 4 C0 HTp0.

We will complete the proof by showing that (A.1) and (A.2) produce a contradiction.

ΠΩ(L, β)

slope β

6

-1

22

33

4

4

55

66

77

8

8

99

10

Figure IIThe polygon ΠΩ(L, β) determined by a L-truncation

If β ∈ R \Q, the polygon ΠΩ(L, β) converges to Ω as L→∞Pick coprimes p 6= q such that p

q ∼ β /Dilate MΩ,β and approximate it by Zpq

Indeed, according to Dirichlet’s diophantine approximation, since β is irrational wemay find infinitely many coprime integers p, q so that |β − p/q| < 1/q2. Denote byI the set of such pairs of coprime integers and pick (p, q) ∈ I. On the other handby dilation-invariance of the Lp0

-operator norm of TMΩ,β, (A.1) implies

(A.3)

∥∥TMp,qΩ,β

: Lp0(R)→ Lp0(R)∥∥ ≤ 2 C0 HTp0

for Mp,qΩ,β(s) = MΩ,β

( √p2+q2

L0

√1+β2

s)

1[0,L0](s).

Divide the segment in γ running from the origin to the point (L0, βL0) into pqequidistributed points. Formally, we identify this segment with the torus T and the


set of points(L0k

pq, β

L0k

pq

): 0 ≤ k ≤ pq − 1

'k/pq : 0 ≤ k ≤ pq − 1

with the cyclic group Zpq. According to (A.3) and de Leeuw’s restriction∥∥T(Mp,q

Ω,β)|Zpq: Lp0

(Zpq)→ Lp0(Zpq)

∥∥ ≤ 2 C0 HTp0.

Since p and q are coprime, we may consider the group isomorphism

Λ : Zpq 3k

pq7→(kp,k

q

)∈ Zp × Zq,

where Zp × Zq is viewed as a lattice of [0, 1]2. It is clear that Λ extends to anisometry on Lp0

of the corresponding dual groups (still denoted by Λ) and weobtain

(A.4)∥∥Tmp,q : Lp0(Zp × Zq)→ Lp0(Zp × Zq)

∥∥ ≤ 2 C0 HTp0 ,

where mp,q(s1, s2) = Mp,qΩ,β(L0Λ−1(s1, s2)). Given 0 ≤ k1 ≤ p−1 and 0 ≤ k2 ≤ q−1

let k = k(k1, k2) be the only integer 0 ≤ k ≤ pq − 1 satisfying that kmod p = k1

and kmod q = k2. Then, we can write

mp,q

(k1

p,k2

q

)= Mp,q

Ω,β

(L0k

pq

)= MΩ,β

( √p2 + q2

pq√

1 + β2k)

1[0,L0]

(L0k

pq

)= MΩ,β(Lp,q)

( √p2 + q2

pq√

1 + β2k)

with Lp,q =

√p2 + q2√1 + β2

.

Letting eβ and e pq

be the unit vectors in the directions of γ and ( 1p ,

1q ) respectively∣∣∣√p2 + q2

pqk

(1, β)√1 + β2

−(kp,k

q

)∣∣∣ =∣∣∣√p2 + q2

pqkeβ −

√p2 + q2

pqke p

q

∣∣∣≤

√p2 + q2

∣∣eβ − e pq

∣∣ . √p2 + q2

q2.

1

q

since we may assume with no loss of generality that β < 1 and p < q. We obtain

mp,q

(kp,k

q

)= 1Ω

((kp,k

q

)+ α(k) + Z2

)= 1ΠΩ(Lp,q,β)

((kp,k

q

)+ α(k) + Z2

)with |α(k)| . 1

q.

We deduce that there must exist a small perturbation Ω(p, q) of Ω so that

mp,q = 1Ω(p,q)|Zp×Zqand Ω(p, q)→ Ω uniformly as p, q →∞.

By considering the symbol mp,q = 1Ω(p,q) : T2 → C for (p, q) ∈ I, we get a sequenceof symbols which converges uniformly to 1Ω and satisfy the uniform estimate below

sup(p,q)∈I

∥∥T(mp,q)|Zp×Zq: Lp0(Zp × Zq)→ Lp0(Zp × Zq)

∥∥ ≤ 2 C0 HTp0 .

By the lattice approximation result obtained in Remark 3.3, this would imply that1Ω yields an Lp0 -bounded Fourier multiplier in T2, and also in R2 by standardperiodization and Hilbert transform truncation. This is a contradiction since Ωadmits Kakeya sets of directions. The proof is complete.


Remark A.2. This result also holds in higher-dimensions by using cocycles intohigher-dimensional spaces, essentially the same argument applies. On the otherhand, when β ∈ R\Q and Ω is a polytope with infinitely many faces not admittingKakeya sets of directions, the conjecture is that such Ω should define a Lp-boundedFourier multiplier (1 < p < ∞) so that we may argue as we did for polyhedronswith finitely many faces. In dimension 2 this is supported by the results in [2, 9] asfor higher-dimensions by [43].

B. Noncommutative Jodeit theorems

Jodeit’s theorem [29] provides another approach to de Leeuw’s compactificationby looking at extensions of Fourier multipliers. He proved that any Lp-boundedFourier multiplier on Zn is the restriction of a Lp-bounded Fourier multiplier onRn. To be more precise, define

Mqp (G) =

m : G→ C

∣∣Tm : Lp(G)→ Lq(G)

for 1 ≤ p ≤ q ≤ ∞. One of the results in [29] is that there is a bounded linearmap φ : Mq

p (Zn) → Mqp (Rn) so that the restriction of φ(m) to Zn is m. When

n = 1, the symbol φ(m) = m is just the multiplier given by the piecewise linearextension m = 1[− 1

2 ,12 ] ∗m ∗ 1[− 1

2 ,12 ] of m. Then the ADS property readily gives

compactification but one looses on the norm by some constant depending on n.

This question of extending multipliers from a subgroup makes sense for generalLCA groups and suits in our framework. A commutative solution was providedby Figa-Talamanca and Gaudry in [17] by extending Jodeit’s result to arbitrarydiscrete subgroups Γ of LCA groups G. Given any such pair, they construct acontractive map φ : Mq

p (Γ)→Mqp (G) so that φ(m) = m with m = ∆∗m∗∆ where

∆ is a positive definite function with small support relative to Γ. This is not theexact analogue of Jodeit’s result (as ∆ = 1[− 1

2 ,12 ] ∗ 1[− 1

2 ,12 ]) but one gains on the

constants. Shortly after, Cowling [11] generalized it to all pairs H ⊂ G where H isclosed but not open, G LCA and m ∈ Cc(H). In the same paper, he also looked atperiodization. The underlying idea is to use suitably the disintegration theory andwith that respect are of commutative nature.

If we restrict ourselves only to discrete subgroups, such a result would perfectlyfit in our framework. In full generality, we yet do not have the right tools toextend Fourier multipliers. However, for the completely bounded ones, we can usetransference from Schur multipliers as in Section 8. Indeed, the latter are muchmore flexible and it is proved in [36, Lemma 2.6] that a Jodeit’s theorem for themis elementary. More precisely, if Γ ⊂ G is a lattice with a symmetric fundamentaldomain X and m : Γ → C is a cb-bounded Schur multipliers on Sp(`2(Γ)), thenm = 1X ∗m ∗ 1X is a cb-bounded Schur multiplier on Sp(L2(G)). In particular weobtain the following extension result.

Theorem B.1. Let Γ ⊂ G be a lattice in an amenable locally compact group G witha symmetric fundamental domain X. For any m : Γ→ C with m = 1X ∗m ∗ 1X∥∥Tm : Lp(G)→ Lp(G)

∥∥cb≤∥∥Tm : Lp(Γ)→ Lp(Γ)

∥∥cb.

In particular, the cb-bounded version of Jodeit’s theorem holds with constant 1.


Acknowledgement. M. Caspers is partially supported by the grant SFB 878“Groups, geometry and actions”; J. Parcet and M. Perrin are partially supportedby the ERC StG-256997-CZOSQP (UE) and ICMAT Severo Ochoa SEV-2011-0087

(Spain); and E. Ricard by ANR-2011-BS01-008-11 (France).

References

[1] Cedric Arhancet, Unconditionality, Fourier multipliers and Schur multipliers, Colloq. Math.

127 (2012), no. 1, 17–37.

[2] Michael Bateman, Kakeya sets and directional maximal operators in the plane, Duke Math.J. 147 (2009), no. 1, 55–77.

[3] Joran Bergh and Jorgen Lofstrom, Interpolation spaces. An introduction, Springer-Verlag,

Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.[4] Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag,

New York, 1997.[5] Marek Bozejko and Gero Fendler, Herz-Schur multipliers and completely bounded multipliers

of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2,

297–302.[6] Alberto P. Calderon, Ergodic theory and translation-invariant operators, Proc. Nat. Acad.

Sci. U.S.A. 59 (1968), 349–353.

[7] Martijn Caspers and Mikael de la Salle, Schur and Fourier multipliers of an amenablegroup acting on non-commutative Lp-spaces, To appear in Trans. Amer. Math. Soc. ArXiv:

1303.0135, 2014.

[8] Ronald R. Coifman and Guido Weiss, Transference methods in analysis, American Math-ematical Society, Providence, R.I., 1976, Conference Board of the Mathematical Sciences

Regional Conference Series in Mathematics, No. 31.

[9] Antonio Cordoba and Robert Fefferman, On the equivalence between the boundedness ofcertain classes of maximal and multiplier operators in Fourier analysis, Proc. Natl. Acad.

Sci. USA 74 (1977), no. 2, 423–425.[10] Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat.

Cuyana 1 (1955), 105–167 (1956).

[11] Michael Cowling, Extension of Fourier Lp − Lq multipliers, Trans. Amer. Math. Soc. 213(1975), 1–33.

[12] Michael Cowling and Uffe Haagerup, Completely bounded multipliers of the Fourier algebra

of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549.[13] Karel de Leeuw, On Lp multipliers, Ann. of Math. (2) 81 (1965), 364–379.

[14] Antoine Derighetti, A property of Bp(G). Applications to convolution operators, J. Funct.

Anal. 256 (2009), no. 3, 928–939.[15] A. H. Dooley and G. I. Gaudry, An extension of de Leeuw’s theorem to the n-dimensional

rotation group, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 111–135.

[16] Charles Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336.[17] Alessandro Figa-Talamanca and Garth I. Gaudry, Extensions of multipliers, Boll. Un. Mat.

Ital. (4) 3 (1970), 1003–1014.

[18] Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics,CRC Press, Boca Raton, FL, 1995.

[19] , Real analysis, second ed., Pure and Applied Mathematics (New York), John Wiley &Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience

Publication.

[20] Stanis law Goldstein and J. Martin Lindsay, Markov semigroups KMS-symmetric for a weight,Math. Ann. 313 (1999), no. 1, 39–67.

[21] Loukas Grafakos, Modern Fourier analysis, second ed., Graduate Texts in Mathematics, vol.250, Springer, New York, 2009.

[22] Siegfried Grosser and Martin Moskowitz, On central topological groups, Trans. Amer. Math.

Soc. 127 (1967), 317–340.[23] Uffe Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approximation

property, Invent. Math. 50 (1978/79), no. 3, 279–293.


[24] Uffe Haagerup, Marius Junge, and Quanhua Xu, A reduction method for noncommutative

Lp-spaces and applications, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2125–2165.

[25] Asma Harcharras, Fourier analysis, Schur multipliers on Sp and non-commutative Λ(p)-sets,Studia Math. 137 (1999), no. 3, 203–260.

[26] Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3,

91–123.[27] Michel Hilsum, Les espaces Lp d’une algebre de von Neumann definies par la derivee spatiale,

J. Funct. Anal. 40 (1981), no. 2, 151–169.

[28] Satoru Igari, Lectures on fourier series in several variables, University of Wisconsin (1968).[29] Max Jodeit, Jr., Restrictions and extensions of Fourier multipliers, Studia Math. 34 (1970),

215–226.

[30] Marius Junge, Fubini’s theorem for ultraproducts of noncommmutative Lp-spaces, Canad. J.Math. 56 (2004), no. 5, 983–1021.

[31] Marius Junge and Tao Mei, Noncommutative Riesz transforms—a probabilistic approach,Amer. J. Math. 132 (2010), no. 3, 611–680.

[32] Marius Junge, Tao Mei, and Javier Parcet, Smooth Fourier multipliers in group von Neumann

algebras, ArXiv: 1010.5320, Preprint, 2012.[33] , Noncommutative Riesz transforms — dimension free bounds and Fourier multipli-

ers, Preprint, 2014.

[34] Marius Junge and David Sherman, Noncommutative Lp modules, J. Operator Theory 53(2005), no. 1, 3–34.

[35] Hideki Kosaki, On the continuity of the map ϕ → |ϕ| from the predual of a W ∗-algebra, J.

Funct. Anal. 59 (1984), no. 1, 123–131. MR 763779 (86c:46072)[36] Vincent Lafforgue and Mikael de la Salle, Noncommutative Lp-spaces without the completely

bounded approximation property, Duke Math. J. 160 (2011), no. 1, 71–116.[37] Vladimir Lebedev and Alexander Olevskiı, Idempotents of Fourier multiplier algebra, Geom.

Funct. Anal. 4 (1994), no. 5, 539–544.

[38] Viktor Losert, A characterization of SIN-groups, Math. Ann. 273 (1985), no. 1, 81–88.[39] Gerd Mockenhaupt and Werner J. Ricker, Idempotent multipliers for Lp(R), Arch. Math.

(Basel) 74 (2000), no. 1, 61–65.

[40] Stefan Neuwirth and Eric Ricard, Transfer of Fourier multipliers into Schur multipliers and

sumsets in a discrete group, Canad. J. Math. 63 (2011), no. 5, 1161–1187.

[41] Javier Parcet and Gilles Pisier, Non-commutative Khintchine type inequalities associated withfree groups, Indiana Univ. Math. J. 54 (2005), no. 2, 531–556.

[42] Javier Parcet and Keith Rogers, Twisted Hilbert transforms vs Kakeya sets of directions, To

appear in J. Reine Angew. Math. Arxiv: 1207.1992, 2012.[43] Javier Parcet and Keith M. Rogers, Differentiation of integrals in higher dimensions, Proc.

Natl. Acad. Sci. USA 110 (2013), no. 13, 4941–4944.[44] Gert K. Pedersen, C∗-algebras and their automorphism groups, London Mathematical Society

Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-

New York, 1979.[45] Gilles Pisier, Non-commutative vector valued Lp-spaces and completely p-summing maps,

Asterisque (1998), no. 247, vi+131.

[46] , Introduction to operator space theory, London Mathematical Society Lecture NoteSeries, vol. 294, Cambridge University Press, Cambridge, 2003.

[47] Gilles Pisier and Quanhua Xu, Non-commutative Lp-spaces, Handbook of the geometry of

Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.[48] Sadahiro Saeki, Translation invariant operators on groups, Tohoku Math. J. (2) 22 (1970),

409–419.[49] Shashi M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180,

Springer-Verlag, New York, 1998.

[50] Masamichi Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sci-ences, vol. 125, Springer-Verlag, Berlin, 2003, Operator Algebras and Non-commutative Ge-

ometry, 6.[51] Marianne Terp, Lp spaces associated with von neumann algebras, Notes, Københavns Uni-

versitets Matematiske Institut (1981).[52] , Interpolation spaces between a von Neumann algebra and its predual, J. Operator

Theory 8 (1982), no. 2, 327–360.


[53] Nicolas Th. Varopoulos, Laurent Saloff-Coste, and Thierry Coulhon, Analysis and geom-

etry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press,

Cambridge, 1992.[54] Norman J. Weiss, A multiplier theorem for SU(n), Proc. Amer. Math. Soc. 59 (1976), no. 2,

366–370.

Martijn CaspersFachbereich Mathematik und Informatik

Westfalische Wilhelmsuniversitat MunsterEinsteinstrasse 62, 48149 Munster, Germany

[email protected]

Javier ParcetInstituto de Ciencias Matematicas

CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones Cientıficas

C/ Nicolas Cabrera 13-15. 28049, Madrid. [email protected]

Mathilde PerrinInstituto de Ciencias Matematicas

CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones Cientıficas

C/ Nicolas Cabrera 13-15. 28049, Madrid. [email protected]

Eric RicardLaboratoire de Mathematiques Nicolas Oresme

Universite de Caen Basse-Normandie14032 Caen Cedex, [email protected]

ICMAT€¦ · NONCOMMUTATIVE DE LEEUW THEOREMS MARTIJN CASPERS, JAVIER PARCET MATHILDE PERRIN AND...

Documents

Transcript of ICMAT€¦ · NONCOMMUTATIVE DE LEEUW THEOREMS MARTIJN CASPERS, JAVIER PARCET MATHILDE PERRIN AND...